multiplicity results for the scalar curvature equation abstract this paper is devoted to the study of positive radial solutions of the scalar curvature equation i e u x k x u 1 x 0 where 2 n n 2 and we assume that k x k x e and k r c 1 is bounded and e 0 is small it is known that we have at least a ground state with fast decay for each positive critical point of k for e small enough in fact if the critical point k r 0 is unique and it is a maximum we also have uniqueness surprisingly we show that if k r 0 is a minimum we have an arbitrarily large number of ground states with fast decay the results are obtained using fowler transformation and developing a dynamical approach inspired by melnikov theory we emphasize that the presence of subharmonic solutions arising from zeroes of melnikov functions has not appeared previously as far as we are aware

interference alignment with asymmetric complex signaling settling the host madsen nosratinia conjecture it has been conjectured by ho madsen and nosratinia that complex gaussian interference channels with constant channel coefficients have only one degree of freedom regardless of the number of users while several examples are known of constant channels that achieve more than 1 degree of freedom these special cases only span a subset of measure zero in other words for almost all channel coefficient values it is not known if more than 1 degree of freedom is achievable in this paper we settle the host madsen nosratinia conjecture in the negative we show that at least 1 2 degrees of freedom are achievable for all values of complex channel coefficients except for a subset of measure zero for the class of linear beamforming and interference alignment schemes considered in this paper it is also shown that 1 2 is the maximum number of degrees of freedom achievable on the complex gaussian 3 user interference channel with constant channel coefficients for almost all values of channel coefficients to establish the achievability of 1 2 degrees of freedom we use the novel idea of asymmetric complex signaling i e the inputs are chosen to be complex but not circularly symmetric it is shown that unlike gaussian point to point multiple access and broadcast channels where circularly symmetric complex gaussian inputs are optimal for interference channels optimal inputs are in general asymmetric with asymmetric complex signaling we also show that the 2 user complex gaussian x channel with constant channel coefficients achieves the outer bound of 4 3 degrees of freedom i e the assumption of time variations frequency selectivity used in prior work to establish the same result is not needed

on the topology and index of minimal surfaces we show that for an immersed two sided minimal surface in r 3 there is a lower bound on the index depending on the genus and number of ends using this we show the nonexistence of an embedded minimal surface in r 3 of index 2 as conjectured by choe cho90 moreover we show that the index of an immersed two sided minimal surface with embedded ends is bounded from above and below by a linear function of the total curvature of the surface

the topology and geometry of embedded surfaces of constant mean curvature where ai p and a2 p are the principal curvatures of m at p when h is constant m is called a surface of constant mean curvature a surface is said to have finite type if it is homeomorphic to a closed surface with a finite number of points removed an important problem in classical differential geometry is the classification of properly embedded finite type surfaces m of constant mean curvature in r if m is a closed embedded surface of constant mean curvature then it follows from alexandrov 1 that m must be a round sphere the classical examples of properly embedded surfaces with zero mean curvature are the plane the helicoid and the catenoid surfaces of zero mean curvature are usually called minimal surfaces the remaining classical examples of properly embedded surfaces of constant mean curvature were found by delaunay 4 the delaunay surfaces are surfaces of revolution recently hoffman and meeks 6 7 have found examples of properly embedded minimal surfaces which are homeomorphic to closed surfaces of positive genus with 3 points removed callahan hoffman and meeks 3 have found other examples with more ends an annular end e of properly embedded surface in r 3 is a properly embedded annulus e in m where e is homeomorphic to s x 0 1 when m has finite type then every end of m is annular hoffman and meeks have developed a theory to deal with global problems concerning the geometry of properly embedded minimal surfaces m and in particular they show that most annular ends of m converge at infinity in r to a flat plane or to the end of a catenoid recently n kapouleas 8 in his thesis has shown that for every positive integer k 2 there exists a properly embedded surface m of finite type with nonzero mean curvature and with k ends he also has constructed highergenus examples as in the case of minimal surfaces the annular ends of a properly embedded surface of nonzero constant mean curvature have a special geometry and play an important role in global theorems

embedded minimal surfaces with an infinite number of ends we prove the existence of an infinite family of periodic properly embedded minimal surfaces with an infinite number of annular ends 25 refs 29 figs

on codimension one foliations of constant mean curvature a codimension one foliation of a riemannian manifold m g is said to be of constant mean curvature resp minimal if each leaf of is a hypersurface of constant mean curvature resp minimal recently foliations of constant mean curvature are studied by some differential geometers 1 i 2 i 8 on the other hand sullivan 1 12 characterized minimal foliations in terms of homology and walczak 14 15 recently tried to find an extension of sullivan s result by considering which function can become a mean curvature function of a given foliation unfortunately the proof in 1 15 has some gaps in this paper we first give a proof of walczak s theorem for the case when m is oriented and then in w 3 by using a similar argument we give a sufficient condition for to be of constant mean curvature this enables us to have various examples of constant mean curvature foliations in fact starting with an arbitrary codimension one foliation we can always find a constant mean curvature foliation concordant with the given one this is done in w 4 the author is grateful to professor g cairns who gave me the first example of a constant mean curvature foliation which can not be minimal the author is also grateful to the hospitality of the mathematical institute university of k61n especially to professor dombrowski and the members of his seminar

a costa hoffman meeks type surface in mathbb h 2 times mathbb r we show the existence in the space 2 x of a family of embedded minimal surfaces of genus 1 k and finite total extrinsic curvature with two catenoidal type ends and one middle planar end the proof is based on a gluing procedure

complete embedded minimal surfaces of finite total curvature with planar ends of smallest possible order using a method by traizet j differ geom 60 103 153 2002 which reduces the construction of minimal surfaces via the weierstras theorem and the implicit function theorem to solving algebraic equations in several complex variables we will show the existence of complete embedded minimal surfaces of finite total curvature with planar ends of least possible order

an updating algorithm for subspace tracking in certain signal processing applications it is required to compute the null space of a matrix whose rows are samples of a signal with p components the usual tool for doing this is the singular value decomposition however the singular value decomposition has the drawback that it requires o p sup 3 operations to recompute when a new sample arrives it is shown that a different decomposition called the urv decomposition is equally effective in exhibiting the null space and can be updated in o p sup 2 time the updating technique can be run on a linear array of p processors in o p time

on feasibility of interference alignment in mimo interference networks we explore the feasibility of interference alignment in signal vector space based only on beamforming for k user mimo interference channels our main contribution is to relate the feasibility issue to the problem of determining the solvability of a multivariate polynomial system which is considered extensively in algebraic geometry it is well known e g from bezout s theorem that generic polynomial systems are solvable if and only if the number of equations does not exceed the number of variables following this intuition we classify signal space interference alignment problems as either proper or improper based on the number of equations and variables rigorous connections between feasible and proper systems are made through bernshtein s theorem for the case where each transmitter uses only one beamforming vector the multibeam case introduces dependencies among the coefficients of a polynomial system so that the system is no longer generic in the sense required by both theorems in this case we show that the connection between feasible and proper systems can be further strengthened since the equivalency between feasible and proper systems does not always hold by including standard information theoretic outer bounds in the feasibility analysis

derivatives and perturbations of eigenvectors for a matrix bf a z whose entries are complex valued functions of a complex variable z results are presented concerning derivatives of an eigenvector bf x z of bf a z associated with a simple eigenvalue lambda z when bf x z is restricted to satisfy a constraint of the form sigma bf x z 1 where sigma is a rather arbitrary scaling function the differentiation formulas lead to a new approach for analyzing the sensitivity of an eigenvector under small perturbations in the underlying matrix finally an application is given which concerns the analysis of a finite markov chain subject to perturbations in the transition probabilities

complete embedded minimal surfaces of finite total curvature we will survey what is known about minimal surfaces s r 3 which are complete embedded and have finite total curvature int s k da infty the only classically known examples of such surfaces were the plane and the catenoid the discovery by costa 14 15 early in the last decade of a new example that proved to be embedded sparked a great deal of research in this area many new examples have been found even families of them as will be described below the central question has been transformed from whether or not there are any examples except surfaces of rotation to one of understanding the structure of the space of examples

embedded minimal ends of finite type we prove that the end of a complete embedded minimal surface in r 3 with infinite total curvature and finite type has an explicit weierstrass representation that only depends on a holomorphic function that vanishes at the puncture reciprocally any choice of such an analytic function gives rise to a properly embedded minimal end e provided that it solves the corresponding period problem furthermore if the flux along the boundary vanishes then the end is c 0 asymptotic to a helicoid we apply these results to proving that any complete embedded one ended minimal surface of finite type and infinite total curvature is asymptotic to a helicoid and we characterize the helicoid as the only simply connected complete embedded minimal surface of finite type in r 3

uniqueness of finite total curvatures and the structure of radial solutions for nonlinear elliptic equations in this article we are concerned with the semilinear elliptic equation where n 2 p 1 and k lxl 0 in r n the correspondence between the initial values of regularly positive radial solutions of the above equation and the associated finite total curvatures will be derived in addition we also conduct the zeros of radial solutions in terms of the initial data under specific conditions on k and p furthermore based on the pohozaev identity and openness for the regions of initial data corresponding to certain types of solutions we obtain the whole structure of radial solutions depending on various situations

stationary isothermic surfaces and uniformly dense domains we establish a relationship between stationary isothermic surfaces and uniformly dense domains a stationary isothermic surface is a level surface of temperature which does not evolve with time a domain in the n dimensional euclidean space r n is said to be uniformly dense in a surface r n of codimension 1 if for every small r 0 the volume of the intersection of with a ball of radius r and center x does not depend on x for x we prove that the boundary of every uniformly dense domain which is bounded or whose complement is bounded must be a sphere we then examine a uniformly dense domain with unbounded boundary and we show that the principal curvatures of satisfy certain identities the case in which the surface coincides with is particularly interesting in fact we show that if the boundary of a uniformly dense domain is connected then i if n 2 it must be either a circle or a straight line and ii if n 3 it must be either a sphere a spherical cylinder or a minimal surface we conclude with a discussion on uniformly dense domains whose boundary is a minimal surface

the approximate capacity of the many to one and one to many gaussian interference channels recently etkin tse and wang found the capacity region of the two user gaussian interference channel to within 1 bit s hz a natural goal is to apply this approach to the gaussian interference channel with an arbitrary number of users we make progress towards this goal by finding the capacity region of the many to one and one to many gaussian interference channels to within a constant number of bits the result makes use of a deterministic model to provide insight into the gaussian channel the deterministic model makes explicit the dimension of signal level a central theme emerges the use of lattice codes for alignment of interfering signals on the signal level

nodal solutions for supercritical laplace equations in this paper we study radial solutions for the following equation delta u x f u x x 0 where x in mathbb r n n 2 f is subcritical for r small and u large and supercritical for r large and u small with respect to the sobolev critical exponent 2 frac 2n n 2 the solutions are classified and characterized by their asymptotic behaviour and nodal properties in an appropriate super linear setting we give an asymptotic condition sufficient to guarantee the existence of at least one ground state with fast decay with exactly j zeroes for any j 0 under the same assumptions we also find uncountably many ground states with slow decay singular ground states with fast decay and singular ground states with slow decay all of them with exactly j zeroes our approach based on fowler transformation and invariant manifold theory enables us to deal with a wide family of potentials allowing spatial inhomogeneity and a quite general dependence on u in particular for the matukuma type potential we show a kind of structural stability

degrees of freedom of the k user m times n mimo interference channel we provide inner bound and outer bound for the total number of degrees of freedom of the k user multiple input multiple output mimo gaussian interference channel with m antennas at each transmitter and n antennas at each receiver if the channel coefficients are time varying and drawn from a continuous distribution the bounds are tight when the ratio max m n min m n r is equal to an integer for this case we show that the total number of degrees of freedom is equal to min m n k if k r and min m n r r 1 k if k r achievability is based on interference alignment we also provide examples where using interference alignment combined with zero forcing can achieve more degrees of freedom than merely zero forcing for some mimo interference channels with constant channel coefficients

an embedded genus one helicoid there exists a properly embedded minimal surface of genus one with a single end asymptotic to the end of the helicoid this genus one helicoid is constructed as the limit of a continuous one parameter family of screw motion invariant minimal surfaces also asymptotic to the helicoid that have genus equal to one in the quotient

deformations of complete minimal surfaces a notion of deformation is defined and studied for complete minimal surfaces in r3 and r3 c c a group of translations the catenoid enneper s surface and the surface of meeks jorge modelled on a 3 punctured sphere are shown to be isolated minimal surfaces of total curvature 4sr in r3 z and r3 z2 are studied it is proved that the helicoid and scherk s surface are isolated under periodic perturbations let m be a submanifold of a riemannian manifold n and let te m be a tubular neighborhood of m in n of radius e a c1 e variation of m in n is a submanifold m1 c te m which is a graph over m and is e c1 close to m this means m1 is pointwise e close to m in each fibre of te m and the tangent planes as well we are interested in complete minimal submanifolds m of n abbreviated c m s and c1 e variations which are also c m s s and henceforth we always assume m and m1 are c m s s we say m is isolated if for some e 0 the only e c1 variations of m differ from m by an ambient isometry of n in this paper we shall study this question when n is r3 or a translation space r3 modulo a group of translations a flat plane in r3is isolated this follows immediately from bernstein s theorem a function of two variables on r2 whose graph is a c m s is linear we shall prove enneper s surface the catenoid and a certain 3 punctured sphere discovered by meeks and jorge are isolated in r3 we prove the helicoid and scherk s surface are isolated under periodic deformations i e the helicoid of total curvature 4sr in r3 modulo one translation is isolated in this translation space and scherk s surface of total curvature 4sr is isolated in r3 modulo two translations this scherk surface is conformally a 4 punctured sphere and the helicoid a 2 punctured sphere we shall study minimal submanifolds of translation spaces many of the tech niques developed by r osserman can be adapted to this context and yield information about periodic minimal surfaces in r3 for example we classify c m s s vf total curvature 4sr in translation space in r3 osserman has proved the catenoid and enneper s surface are the only c m s s of total curvature 4sr we obtain an analogous classification in r3 g in vi we describe these surfaces for g z we study c m s s m of finite total curvature in translation spaces we prove a deformation i e an e c1 variation m1 of a finite total curvature c m s in a translation space is also of finite total curvature and is conformally a compact

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

minimal hypersurfaces with bounded index we prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index and volume if the ambient dimension is greater than three in a riemannian manifold m n g 3 le n le 7 can degenerate loosely speaking our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces up to controlled errors several compactness finiteness theorems follow from our local picture

degrees of freedom region of the mimo x channel we provide achievability as well as converse results for the degrees of freedom region of a multiple input multiple output mimo x channel i e a system with two transmitters two receivers each equipped with multiple antennas where independent messages need to be conveyed over fixed channels from each transmitter to each receiver the inner and outer bounds on the degrees of freedom region are tight whenever integer degrees of freedom are optimal for each message with m 1 antennas at each node we find that the total sum rate degrees of freedom are bounded above and below as 1 les eta x les 4 3 if m 1 and channel matrices are nondegenerate then the precise degrees of freedom eta x 4 3 m thus the mimo x channel has noninteger degrees of freedom when m is not a multiple of 3 simple zero forcing without dirty paper encoding or successive decoding suffices to achieve the 4 3 m degrees of freedom if the channels vary with time frequency then the channel with single antennas m 1 at all nodes has exactly 4 3 degrees of freedom the key idea for the achievability of the degrees of freedom is interference alignment i e signal spaces are aligned at receivers where they constitute interference while they are separable at receivers where they are desired we also explore the increase in degrees of freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio

noniterative subspace tracking a rank one spherical subspace update that is appropriate for subspace based methods like music and minimum norm is introduced this noniterative highly parallel numerically stabilized subspace update is closely related to rank one eigenstructure updating however a rank one subspace update involves less computation than simple rank one correlation accumulation moreover the frequency tracking capabilities of the noniterative subspace update are virtually identical to and in some case more robust than the more computationally expensive eigen based methods

entire solutions of the allen cahn equation and complete embedded minimal surfaces of finite total curvature in r 3 we consider minimal surfaces m which are complete embedded and have finite total curvature in mathbb r 3 and bounded entire solutions with finite morse index of the allen cahn equation delta u f u 0 in mathbb r 3 here f w with w bi stable and balanced for instance w u frac 1 4 1 u 2 2 we assume that m has m ge 2 ends and additionally that m is non degenerate in the sense that its bounded jacobi fields are all originated from rigid motions this is known for instance for a catenoid and for the costa hoffman meeks surface of any genus we prove that for any small alpha gt 0 the allen cahn equation has a family of bounded solutions depending on m 1 parameters distinct from rigid motions whose level sets are embedded surfaces lying close to the blown up surface m alpha alpha 1 m with ends possibly diverging logarithmically from m alpha we prove that these solutions are l infty non degenerate up to rigid motions and find that their morse index coincides with the index of the minimal surface our construction suggests parallels of de giorgi conjecture for general bounded solutions of finite morse index

higher genus riemann minimal surfaces riemann surfaces constitute a one parameter family of embedded minimal surfaces which are periodic and have infinitely many horizontal planar ends the surfaces in this family are foliated by circles or straight lines in this paper we prove the existence of a one parameter family of embeded minimal surfaces which have infinitely many horizontal planar ends and have genus k for k 1 37 riemann surfaces as their flux is nearly vertical can be understood as a sequence of parallel planes connected by slightly bent catenoidal neks the surfaces we construct are obtained by replacing one of these catenoidal necks by a member of the family of minimal surfaces discovered by c costa d hoffman and w meek

the asymptotic behavior of properly embedded minimal surfaces of finite topology while advances have been made in recent years in the study of properly embedded minimal surfaces of finite topology in r3 3 5 8 9 11 12 13 23 progress has depended in an essential manner on the special structure of such surfaces with finite total curvature a properly immersed minimal surface with finite total curvature is conformally a compact riemann surface punctured in a finite number of points and its gauss map extends to the compact surface as a meromorphic function 22 in particular all the topological ends are conformally equivalent to a punctured disk and there is a well defined limit tangent plane at each end outside of a sufficiently large compact set such an end is a multisheeted graph over the limit tangent plane and if the end is embedded it is asymptotic to either a plane or a half catenoid see e g schoen 24 for a survey of results on properly immersed surfaces of finite total curvature see 9 1 without the assumption of finite total curvature these results are not true in general as an indication of the relative lack of information consider that it is unknown whether or not the helicoid is the only simply connected minimal surface that is properly embedded and nonplanar a question raised by osserman the helicoid has infinite total curvature and a single annular end if a surface has finite topology then all of its ends are annular any general theory of properly embedded minimal surfaces of finite topology must include an understanding of the behavior of these ends in this paper we prove

the degrees of freedom of the k user gaussian interference channel is discontinuous at rational channel coefficients the degrees of freedom of a k user gaussian interference channel gic has been defined to be the multiple of 1 2 log 2 p at which the maximum sum of achievable rates grows with increasing power p in this paper we establish that the degrees of freedom of three or more user real scalar gics viewed as a function of the channel coefficients is discontinuous at points where all of the coefficients are nonzero rational numbers more specifically for all k 2 we find a class of k user gics that is dense in the gic parameter space for which k 2 degrees of freedom are exactly achievable and we show that the degrees of freedom for any gic with nonzero rational coefficients is strictly smaller than k 2 these results are proved using new connections with number theory and additive combinatorics

existence of positive solutions for a class of semilinear and quasilinear elliptic equations with supercritical case abstract in this paper we consider a class of semilinear elliptic dirichlet problems in a bounded regular domain with cylindrical symmetry involving concave convex nonlinearities with supercritical growth using a new sobolev embedding theorem and variational method we show the existence of two positive solutions of the problem additionally we study the quasilinear elliptic equation and obtain a similar result

some remarks on deformations of minimal surfaces we consider complete minimal surfaces c m s s in r3 and their deformations m1 is an s deformation of mo if m1 is a graph over mo in an e tubular neighborhood of mo and m1 is e c1 close to mo a minimal surface m is isolated if all c m s s which are sufficiently small deformations of m are congruent to m in this paper we construct an example of a nonisolated c m s it is modelled on a 4 punctured sphere and is of finite total curvature on the other hand we prove that a c m s discovered by meeks and jorge modelled on the sphere punctured at the fourth roots of unity is isolated introduction we consider complete minimal surfaces c m s s in r3 and their deformations m1 in an s deformation of mo if m1 is a graph over mo is an e tubular neighborhood of m1 and m1 is e c1 close to mo a c m s mo is isolated if all minimal surfaces m1 which are sufficiently small deformations of mo are congruent to mo many of the classical minimal surfaces in r3 are known to be isolated 2 however no example was known of a nonisolated minimal surface in this paper we construct such an example it is modelled on a 4 punctured sphere and is of finite total curvature on the other hand we prove that a c m s discovered by meeks and jorge modelled on the sphere punctured at the four roots of unity 1 1 i i is isolated the analogous surface modelled on the sphere punctured at the cube roots of unity was shown to be isolated in 2 the question is raised there of whether deformations of the four puncture case can be realized by deforming the conformal structure i e changing the cross ratio of the four points thus for the meeks jorge example in question the answer is no as we shall see the conformal structure of our example that admits deformations does not change either perhaps the conformal structure never changes by small deformations we wish to thank w meeks and b morin for helpful conversations and greatly simplifying suggestions i a deformable surface let m be a c m s of finite total curvature so that m is conformally equivalent to a compact riemann surface m punctured at a finite number of points an end e of m is said to be bounded if e is a bounded distance from a plane if e corresponds to the puncture p and p is the plane orthogonal to g p g the gauss map then e bounded means e is a bounded distance from p if e is embedded then it can be expressed as the graph of a log r o rl where r is the distance from the origin in p and this holds for r large then e is bounded if and only if a o let 9 be a weierstrass representation of m and 1 x2 x3 the associated analytic differentials on m we know the real periods of each xk are zero on m received by the editors december 2 1983 and in revised form may 1 1984 1980 mathematics subject classification 1985 revision primary 53a10 1986 american mathematical society 0002 9947 86 1 00 25 per page

cyclic interference alignment for full duplex multi antenna cellular networks this paper studies full duplex fd cellular networks in which a base station bs operated in fd mode with multiple antennas supports multiple uplink and downlink users simultaneously in the same wireless channel two typical fd cellular scenarios are considered one with half duplex hd users and the other with fd users along with the fd bs for both the cases a novel constructive method is developed for finding a closed form interference alignment ia solution named cyclic ia the core idea behind this approach is to construct a set of loop equations enabling ia in a cyclic manner so that beamforming vectors are sequentially determined by solving an eigenvalue problem it is shown analytically that the proposed cyclic ia can achieve the optimal sum degrees of freedom dof when the number of user antennas is large enough to meet the derived conditions in particular it is shown that the proposed scheme achieves a twofold dof gain compared with conventional hd cellular networks even in the presence of inter link interference provided the number of users becomes large enough compared with the ratio of the number of bss and user antennas simulation results demonstrate that not only are the analytical dof results valid but under a practical multi cell scenario the proposed cyclic ia offers significant throughput gains depending on the cell radius

constant mean curvature surfaces in m 2 times mathbf r the subject of this paper is properly embedded h surfaces in riemannian three manifolds of the form m 2 x r where m 2 is a complete riemannian surface when m 2 r 2 we are in the classical domain of h surfaces in r 3 in general we will make some assumptions about m 2 in order to prove stronger results or to show the effects of curvature bounds in m 2 on the behavior of h surfaces in m 2 x r

beam tracking for interference alignment in slowly fading mimo interference channels a perturbations approach under a linear framework in this paper the beam design for signal space interference alignment in slowly fading multiuser multiple input multiple output mimo interference channels is considered based on a linear formulation for interference alignment a predictive beam tracking algorithm is proposed using matrix perturbation theory the proposed algorithm based on a mixture of iteration and update computes interference aligning beamforming vectors at the current time by updating the previous beam vectors based on the channel difference between the two time steps during the predictively updating phase and yields significant reduction in computational complexity compared with existing methods recalculating beams at each time step the tracking performance of the algorithm is analyzed in terms of mean square error and sum rate loss between the predictively updating approach and the recalculating approach and the impact of imperfect channel knowledge is also investigated under the state space channel model numerical results show that the proposed algorithm has almost the same performance as non predictive methods in sum rate thus the proposed algorithm provides a very efficient way to realize interference alignment in a realistic slowly fading mimo channel environment

structure of the sets of regular and singular radial solutions for a semilinear elliptic equation abstract this paper is concerned with the structure of the set of radially symmetric solutions for the equation u f u 0 on r n 0 with n 2 here the nonlinear term f is assumed to be a smooth function of u that is positive for u 0 and is equal to 0 for u 0 then any radial solution u u r r x of the equation is shown to be classified into one of several types according to its behavior as r 0 and r under the assumption that f is supercritical for small u 0 and is subcritical for large u 0 we clarify the entire structure of the set of solutions of various types the pohozaev identity plays a crucial role in the investigation of the structure

stable hypersurfaces with constant scalar curvature it is well known that hypersurfaces m n with constant mean curvature in a riemannian manifold overline m n 1 c of constant sectional curvature c are solutions to the variational problem of extremizing the area function for volumepreserving variations

perturbation of null spaces with application to the eigenvalue problem and generalized inverses we consider properties of a null space of an analytically perturbed matrix in particular we obtain taylor expansions for the eigenvectors which constitute a basis for the perturbed null space furthermore we apply these results to the calculation of puiseux expansion of the perturbed eigenvectors in the case of general eigenvalue problem as well as to the calculation of laurent series expansions for the perturbed group inverse and pseudoinverse matrices

some uniqueness and nonexistence theorems for embedded minimal surfaces thanks to the works of callahan costa hoffman karcher meeks rosenberg wei etc see for example 1 2 3 5 8 11 12 16 20 we dispose now of a large number of properly embedded minimal surfaces in the euclidean space ir other than the classical examples all those surfaces can be viewed as minimal surfaces with finite total curvature properly embedded in complete flat three manifolds the most basic invariants associated to a surface of this type are its topology and its periodicity it is an interesting problem to decide if the simplest examples like the catenoid the helicoid the scherk s surfaces or the riemann example can be characterized in terms of some of these invariants in the non periodic case there are two important uniqueness theorems in this direction the first one obtained by schoen 17 characterizes the catenoid as the only complete minimal surface embedded

on updating signal subspaces the authors develop an algorithm for adaptively estimating the noise subspace of a data matrix as is required in signal processing applications employing the signal subspace approach the noise subspace is estimated using a rank revealing qr factorization instead of the more expensive singular value or eigenvalue decompositions using incremental condition estimation to monitor the smallest singular values of triangular matrices the authors can update the rank revealing triangular factorization inexpensively when new rows are added and old rows are deleted experiments demonstrate that the new approach usually requires o n sup 2 work to update an n n matrix and that it accurately tracks the noise subspace

the classification of complete minimal surfaces with total curvature greater than 12 we classify complete orientable minimal surfaces with finite total curvature 87r introduction the classification of complete minimal surfaces with finite total curvature in r3 has been an important problem in the classical differential geometry some basic properties of these surfaces were studied by r osserman see 8 9 who showed the first nontrivial result about this subject concretely he characterized the catenoid and enneper surface as the unique complete orientable minimal surfaces of total curvature 47 however until recent years no more relevant results have been obtained w h meeks 6 gave the classification of nonorientable complete minimal surfaces with total curvature greater than 87 this paper is concerned with the total classification of orientable complete minimal surfaces with total curvature 87 chen and gackstatter 1 discovered the first example of a complete minimal surface properly of genus 1 see theorem 1 the picture of chen gackstatter surface is obtained by joining a handle on enneper s surface this genus one minimal surface has total curvature 87 and no other examples of such surfaces were found so it is expected that no other genus one orientable minimal surface of total curvature 87 does exist in this paper we give a proof of this fact more precisely we prove that chen gackstatter surface is the only genus one orientable complete minimal surface with finite total curvature 87 of course it is not difficult to find genus zero minimal surfaces with total curvature 8i a geometrically interesting example described by jorge and meeks 5 is the trinoid this surface has three embedded catenoid ends moreover its normal vectors at these ends are placed symmetrically in an equator of s2 received by the editors april 27 1990 and in revised form july 20 1990 1980 mathematics subject classification 1985 revision primary 53a10 research partially supported by dgicyt grant ps87 01 15 c03 02

least squares approach to joint beam design for interference alignment in multiuser multi input multi output interference channels in this correspondence the problem of interference alignment for k user time invariant multi input multi output interference channels is considered the necessary and sufficient conditions for interference alignment are converted to a system of linear equations that have dummy variables based on this linear system a new algorithm for beam design for interference alignment is proposed by minimizing the overall interference misalignment the proposed algorithm consists of solving a least squares problem iteratively the convergence of the proposed algorithm is established and its complexity is analyzed the performance of the proposed algorithm is also evaluated numerically it is shown that the proposed algorithm has faster convergence and lower complexity than the previous method with a comparable sum rate performance in the most practical case of two receive antennas

the topology and geometry of embedded surfaces of constant mean curvature si m est une surface proprement plongee dans r 3 avec une courbure moyenne constante non nulle alors m n est pas homeomorphe a une surface fermee avec un seul point elimine soit m une surface proprement plongee dans r 3 a courbure moyenne constante non nulle si m est homeomorphe a une surface fermee avec 2 points elimines alors m est une distance bornee a partir d une ligne droite soit m une surface proprement plongee de courbure moyenne non nulle qui est homeomorphe a une surface fermee avec 3 points elimines alors m reste une distance bornee a partir d un plan

the embedded singly periodic scherk costa surfaces we give a positive answer to m traizet s open question about the existence of complete embedded minimal surfaces with scherk ends without planar geodesics in the singly periodic case these examples get close to an extension of traizet s result concerning asymmetric complete minimal submanifolds of open image in new window with finite total curvature

the space of complete minimal surfaces with finite total curvature as lagrangian submanifold the space m of nondegenerate properly embedded minimal surfaces in r3 with finite total curvature and fixed topology is an analytic lagrangian submanifold of cn where n is the number of ends of the surface in this paper we give two expressions one integral and the other pointwise for the second fundamental form of this submanifold we also consider the compact boundary case and we show that the space of stable nonflat minimal annuli that bound a fixed convex curve in a horizontal plane having a horizontal end of finite total curvature is a locally convex curve in the plane c mathematics subject classification 53a10 53c42

projection approximation subspace tracking subspace estimation plays an important role in a variety of modern signal processing applications we present a new approach for tracking the signal subspace recursively it is based on a novel interpretation of the signal subspace as the solution of a projection like unconstrained minimization problem we show that recursive least squares techniques can be applied to solve this problem by making an appropriate projection approximation the resulting algorithms have a computational complexity of o nr where n is the input vector dimension and r is the number of desired eigencomponents simulation results demonstrate that the tracking capability of these algorithms is similar to and in some cases more robust than the computationally expensive batch eigenvalue decomposition relations of the new algorithms to other subspace tracking methods and numerical issues are also discussed