adding handles to the helicoid there exist two new embedded minimal surfaces asymptotic to the helicoid one is periodic with quotient by orientation preserving translations of genus one the other is nonperiodic of genus one

properly embedded minimal planar domains in 1997 collin proved that any properly embedded minimal surface in r 3 with nite topology and more than one end has

generic mean curvature flow i generic singularities it has long been conjectured that starting at a generic smooth closed embedded surface in r 3 the mean curvature ow remains smooth until it arrives at a singularity in a neighborhood of which the ow looks like concentric spheres or cylinders that is the only singularities of a generic ow are spherical or cylindrical we will address this conjecture here and in a sequel the higher dimensional case will be addressed elsewhere the key to showing this conjecture is to show that shrinking spheres cylinders and planes are the only stable self shrinkers under the mean curvature ow we prove this here in all dimensions an easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away

the singular set of mean curvature flow with generic singularities a mean curvature flow starting from a closed embedded hypersurface in mathbf r n 1 must develop singularities we show that if the flow has only generic singularities then the space time singular set is contained in finitely many compact embedded n 1 dimensional lipschitz submanifolds plus a set of dimension at most n 2 if the initial hypersurface is mean convex then all singularities are generic and the results apply in mathbf r 3 and mathbf r 4 we show that for almost all times the evolving hypersurface is completely smooth and any connected component of the singular set is entirely contained in a time slice for 2 or 3 convex hypersurfaces in all dimensions the same arguments lead to the same conclusion the flow is completely smooth at almost all times and connected components of the singular set are contained in time slices a key technical point is a strong parabolic reifenberg property that we show in all dimensions and for all flows with only generic singularities we also show that the entire flow clears out very rapidly after a generic singularity these results are essentially optimal

bending the helicoid we construct colding minicozzi limit minimal laminations in open domains in mathbb r 3 with the singular set of c 1 convergence being any properly embedded c 1 1 curve by meeks c 1 1 regularity theorem the singular set of convergence of a colding minicozzi limit minimal lamination mathcal l is a locally finite collection s mathcal l of c 1 1 curves that are orthogonal to the leaves of the lamination thus our existence theorem gives a complete answer as to which curves appear as the singular set of a colding minicozzi limit minimal lamination in the case the curve is the unit circle mathbb s 1 1 in the x 1 x 2 plane the classical bjorling theorem produces an infinite sequence of complete minimal annuli h n of finite total curvature which contain the circle the complete minimal surfaces h n contain embedded compact minimal annuli overline h n in closed compact neighborhoods n n of the circle that converge as n to infty to mathbb r 3 x 3 axis in this case we prove that the overline h n converge on compact sets to the foliation of mathbb r 3 x 3 axis by vertical half planes with boundary the x 3 axis and with mathbb s 1 1 as the singular set of c 1 convergence the overline h n have the appearance of highly spinning helicoids with the circle as their axis and are named bent helicoids

the rigidity of embedded constant mean curvature surfaces we study the rigidity of complete embedded constant mean curvature surfaces in r 3 among other things we prove that when such a surface has finite genus then intrinsic isometries of the surface extend to isometries of r 3 or its isometry group contains an index two subgroup of isometries that extend mathematics subject classification primary 53a10 secondary 49q05 53c42

a note on the sample complexity of the er spud algorithm by spielman wang and wright for exact recovery of sparsely used dictionaries we consider the problem of recovering an invertible n n matrix a and a sparse n p random matrix x based on the observation of y ax up to a scaling and permutation of columns of a and rows of x using only elementary tools from the theory of empirical processes we show that a version of the er spud algorithm by spielman wang and wright with high probability recovers a and x exactly provided that p cn log n which is optimal up to the constant c

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

tilting modules and auslander s gorenstein property abstract the notion of generalized tilting modules is introduced over arbitrary rings it is shown that many results on tilting modules over artin algebras are naturally generalized the auslander s gorenstein property for tilting modules is considered over noetherian rings

tame generalized canonical algebras abstract recently a new wide class of finite dimensional algebras whose auslander reiten quivers admit infinite families of faithful standard stable tubes called generalized canonical algebras has been introduced in a skowronski colloq math 90 2001 77 93 we give a complete description of all tame generalized canonical algebras and prove that all these algebras have tame derived categories of bounded complexes

the space of embedded minimal surfaces of fixed genus in a 3 manifold v fixed genus this paper is the fifth and final in a series on embedded minimal surfaces following our earlier papers on disks we prove here two main structure theorems for itnonsimply connected embedded minimal surfaces of any given fixed genus the first of these asserts that any such surface without small necks can be obtained by gluing together two oppositely oriented double spiral staircases the second gives a pair of pants decomposition of any such surface when there are small necks cutting the surface along a collection of short curves after the cutting we are left with graphical pieces that are defined over a disk with either one or two sub disks removed a topological disk with two sub disks removed is called a pair of pants both of these structures occur as different extremes in the two parameter family of minimal surfaces known as the riemann examples

compactness of self shrinkers in r3 with fixed genus abstract we prove the compactness of self shrinkers in r 3 with bounded entropy and fixed genus as a corollary we show that numbers of ends of such surfaces are uniformly bounded by the entropy and genus

inequivalent complexity criteria for free boundary minimal surfaces abstract we obtain a series of results in the global theory of free boundary minimal surfaces which in particular provide a rather complete picture for the way different complexity criteria such as area topology and morse index compare beyond the regime where effective estimates are at disposal

when the kernel of a complete hereditary cotorsion pair is the additive closure of a tilting module abstract in this paper we study when the kernel of a complete hereditary cotorsion pair is the additive closure of a tilting module applications go in three directions the first is to characterize when the little finitistic dimension is finite the second is to obtain equivalent formulations for a wakamatsu tilting module to be a tilting module the third is to give some new characterizations of gorenstein rings

the classification of cmc foliations of bbb r 3 and bbb s 3 with countably many singularities in this paper we generalize our previous local removable singularity theorem for minimal laminations to the case of weak h laminations with h in bbb r constant in a punctured ball of a riemannian three manifold we also obtain a curvature estimate for any weak cmc foliation with possibly varying constant mean curvature from leaf to leaf of a compact riemannian three manifold n with boundary solely in terms of a bound of the absolute sectional curvature of n and of the distance to the boundary of n we then apply these results to classify weak cmc foliations of bbb r 3 and bbb s 3 with a closed countable set of singularities

three circles theorems for schrodinger operators on cylindrical ends and geometric applications we show that for a schrodinger operator with bounded potential on a manifold with cylindrical ends the space of solutions that grows at most exponentially at infinity is finite dimensional and for a dense set of potentials or equivalently for a surface for a fixed potential and a dense set of metrics the constant function 0 is the only solution that vanishes at infinity clearly for general potentials there can be many solutions that vanish at infinity one of the key ingredients in these results is a three circles inequality or log convexity inequality for the sobolev norm of a solution u to a schrodinger equation on a product n 0 t w heren is a closed manifold with a certain spectral gap examples of such n s are all round spheres s n for n 1 and all zoll surfaces finally we discuss some examples arising in geometry of such manifolds and schrodinger operators c

the local picture theorem on the scale of topology we prove a descriptive theorem on the extrinsic geometry of an embedded minimal surface of injectivity radius zero in a homogeneously regular riemannian threemanifold in a certain small intrinsic neighborhood of a point of almost minimal injectivity radius this structure theorem includes a limit object which we call a minimal parking garage structure on r 3 whose theory we also develop

the geometry of minimal surfaces of finite genus i curvature estimates and quasiperiodicity let mathcal m be the space of properly embedded minimal surfaces in 3 with genus zero and two limit ends normalized so that every surface m mathcal m has horizontal limit tangent plane at infinity and the vertical component of its flux equals one we prove that if a sequence m i i mathcal m has the horizontal part of the flux bounded from above then the gaussian curvature of the sequence is uniformly bounded this curvature estimate yields compactness results and the techniques in its proof lead to a number of consequences concerning the geometry of any properly embedded minimal surface in 3 with finite genus and the possible limits through a blowing up process on the scale of curvature of a sequence of properly embedded minimal surfaces with locally bounded genus in a homogeneously regular riemannian 3 manifold

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

multi valued graphs in embedded constant mean curvature disks in this paper we prove that an embedded constant mean curvature disk with gaussian curvature large at a point contains a multi valued graph around that point on the scale of a 2 this generalizes colding and minicozzi s result for minimal surfaces

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

the homological theory of quasi resolving subcategories abstract as a generalization of a resolving subcategory which was introduced by auslander and bridger in 2 the notion of a quasi resolving subcategory of an abelian category is introduced the homological theory on quasi resolving subcategories is studied it is shown that many results in 1 2 4 5 6 8 13 14 17 18 22 25 26 27 are unified and naturally generated

embedded minimal disks proper versus nonproper global versus local we construct a sequence of compact embedded minimal disks in a ball in r 3 with boundaries in the boundary of the ball and where the curvatures blow up only at the center the sequence converges to a limit which is not smooth and not proper if instead the sequence of embedded disks had boundaries in a sequence of balls with radii tending to infinity then we have shown previously that any limit must be smooth and proper

the existence of embedded minimal surfaces and the problem of uniqueness it was in the early nineteen thirties that douglas and radd solved the plateau problem for a rectifiable jordan curve in euclidean space the minimal surfaces that they obtained are realized by conformal harmonic maps from the unit disk in 1948 morrey generalized the theorem of douglas and radd to minimal surfaces in a homogeneously regular riemannian manifold in both cases the solutions are defective as they may possess branch points it was not until 1968 that osserman 24 was able to prove that interior branch points do not exist on the douglas radd morrey solution osserman s theorem in this full generality was proved by alt 2 3 and gulliver 8 if the jordan curve is real analytic gulliver and lesley 9 showed that the solution surface is free of branch points even on its boundary hence at least in this case we know that the douglas radd morrey solution is an immersion and represents a classical surface however the douglas radd morrey solution may in general have selfintersections it was then a classical problem to find conditions on the jordan curve to guarantee that the solution is an embedded disk in 1932 radd was the first to produce a general condition of this type he proved that if the curve can be projected onto a convex curve in the plane then it bounds a unique embedded minimal disk which is a graph over this plane it was not until the seventies that there was further progress on the problem of embeddedness osserman had conjectured that if a jordan curve is extreme that is lies on the boundary of its convex hull then the douglas rad6 solution is embedded gulliver and spruck 10 proved this conjecture under the additional assumption that the absolute total curvature of the jordan curve is less than 4 z right after this work tomi and tromba 32 showed that an extreme curve must bound an embedded minimal disk which need not be stable in 1976 almgren and simon 1 established the existence for such a curve of an embedded stable minimal disk which need not be a douglas radd solution finally in the same year the authors 18 proved that any douglas rad6 solution must be embedded

tilting modules of finite projective dimension tilting modules are well known and are useful in the representation theory of artinian algebras 2 4 6 11 the purpose of this paper is to define tilting modules of finite projective dimension and prove some fundamental results a unital left a module a t over a ring a is said to be a tilting module of projective dimension 0 is an integer if it satisfies the following three conditions 1 a t has a projective resolution

structure theorems for singular minimal laminations in this paper we apply our local removable singularity theorem and local structure theorems for embedded minimal surfaces and minimal laminations in r 3 proven in 16 15 to obtain global structure theorems for certain possibly singular minimal laminations of r 3 we will use theorems 1 3 and theorem 1 6 below in 14 to prove that a complete embedded minimal surface in r 3 with finite genus and a countable number of ends is proper theorem 1 6 will also be applied in 13 to obtain bounds on the index and the topology of complete embedded minimal surfaces of fixed genus and finite topology in r 3

quantitative estimates of the convergence of the empirical covariance matrix in log concave ensembles let k be an isotropic convex body in rn given e 0 how many independent points xi uniformly distributed on k are neededfor the empirical covariance matrix to approximate the identity up to e with overwhelming probability our paper answers this question from 12 more precisely let x rn be a centered random vector with a log concave distribution and with the identity as covariance matrix an example of such a vector x is a random point in an isotropic convex body we show that for any e 0 there existsc e 0 su ch that if n c e n and xi i n are i i d copies of 1 n x then n i 1 xi xi id e with probability larger than 1 exp c n

tilting functors and stable equivalences for selfinjective algebras 1 exta t t 0 2 ext t 0 3 there is a short exact sequence 0 a t t 0 with t t being direct sums of summands of t then putting b end t we call b t a and hom t mod a mod b a tilting triple and a tilting functor respectively tilting functors have been introduced by brenner and butler 6 as a generalization of the bernstein gelfand ponomarev s reflection functors s they and happel and ringel s have proved that we have usually nonhereditary torsion theories f f in mod a and x g in mod b where f xe mod a extj t x 0 and x yemod bi yob t o and the tilting functor and ext t give category equivalences between f and g and between f and x respectively these equivalences give us however no information about indecomposable a modules and b modules which do not belong to the above subcategories f f z and cv the purpose of this paper is to point out that there is a method to enlarge our view by which we can supply the lack of information let us consider trivial extension algebras r a k da and s b k db of a and b by da and db respectively where da hom a k and bdb hom b k are injective cogenerator bimodules in this case r and s are selfinjective more precisely symmetric algebras and mod a and mod b are naturally embedded into projectively stable categories m r and m s then our main theorem states that for any tilting triple

on the uniqueness of the helicoid and enneper s surface in the lorentz minkowski space mathbb r _ 1 3

curvature estimates for surfaces with bounded mean curvature estimates for the norm of the second fundamental form a play a crucial role in studying the geometry of surfaces in r3 in fact when a is bounded the surface cannot bend too sharply in this paper we prove that for an embedded geodesic disk with bounded l2 norm of a a is bounded at interior points provided that the w 1 p norm of its mean curvature is sufficiently small p 2 in doing this we generalize some renowned estimates on a for minimal surfaces

generalized tilting modules with finite injective dimension let r be a left noetherian ring s be a right noetherian ring and r be a generalized tilting module with s end r we mainly consider the conditions such that l id r 1 in this paper and obtain some characterizations for this inequality further when l id r and r id s are finite we provide the conditions such that r and s are injective as a direct application we get some necessary and sufficient conditions for a gorenstein ring to be a qf ring

the space of embedded minimal surfaces of fixed genus in a 3 manifold iv locally simply connected this paper is the fourth in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed 3manifold the key is to understand the structure of an embedded minimal disk in a ball in r3 this was undertaken in cm3 cm4 and the global version of it will be completed here see the discussion around figure 12 for the local case and cm15 for some more details our main results are theorem 0 1 the lamination theorem and theorem 0 2 the one sided curvature estimate the lamination theorem is stated in the global case where the lamination is in fact a foliation the first four papers of this series show that every embedded minimal disk is either a graph of a function or is a double spiral staircase where each staircase is a multivalued graph this is done by showing that if the curvature is large at some point and hence the surface is not a graph then it is a double spiral staircase like the helicoid to prove that such a disk is a double spiral staircase we showed in the first three papers of the series that it is built out of n valued graphs where n is a fixed number in this paper we will deal with how the multi valued graphs fit together and in particular prove regularity of the set of points of large curvature the axis of the double spiral staircase the first theorem is the global version of the statement that every embedded minimal disk is a double spiral staircase

the space of embedded minimal surfaces of fixed genus in a 3 manifold iii planar domains this paper is the third in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed 3 manifold in cm3 cm5 we describe the case where the surfaces are topologically disks on any fixed small scale although the focus of this paper general planar domains is more in line with cm6 we will prove a result here namely corollary iii 3 5 below which is needed in cm5 even for the case of disks roughly speaking there are two main themes in this paper the first is that stability leads to improved curvature estimates this allows us to find large graphical regions these graphical regions lead to two possibilities

non proper helicoid like limits of closed minimal surfaces in 3 manifolds we show that there exists a metric with positive scalar curvature on s 2 s 1 and a sequence of embedded minimal cylinders that converges to a minimal lamination that in a neighborhood of a strictly stable 2 sphere is smooth except at two helicoid like singular ities on the 2 sphere the construction is inspired by a recent example by d hoffman and b white

on modules with trivial self extensions throughout this note all rings will be self basic connected artin algebras over a fixed commutative local artin ring k and all modules will be finitely generated homomorphisms will be written on the opposite side of the scalars by mod a resp a mod we denote the category of all right resp left a modules for an artin algebra a the ordinary duality functor is denoted by d mod a 2 a mod i e d hom f with the minimal injective cogenerator i over k we are interested in the bimodules bta with the following properties

the geometry of minimal surfaces of finite genus ii nonexistence of one limit end examples we demonstrate that a properly embedded minimal surface in 3 with finite genus cannot have one limit end

dictionary learning with few samples and matrix concentration let a be an n times n matrix x be an n times p matrix and y ax a challenging and important problem in data analysis motivated by dictionary learning and other practical problems is to recover both a and x given y under normal circumstances it is clear that this problem is underdetermined however in the case when x is sparse and random spielman et al showed that one can recover both a and x efficiently from y with high probability given that p the number of samples is sufficiently large their method works for p ge c n 2 log 2 n and they conjectured that p ge c n log n suffices the bound n log n is sharp for an obvious information theoretical reason in this paper we show that p ge c n log 4 n suffices matching the conjectural bound up to a polylogarithmic factor the core of our proof is a theorem concerning l 1 concentration of random matrices which is of independent interest our proof of the concentration result is based on two ideas the first is an economical way to apply the union bound the second is a refined version of bernstein s concentration inequality for the sum of independent variables both have nothing to do with random matrices and are applicable in general settings

auslander type conditions and cotorsion pairs we study the properties of rings satisfying auslander type conditions if an artin algebra satisfies the auslander condition that is is an gorenstein artin algebra then we construct two kinds of subcategories which form functorially finite cotorsion pairs

extensions over hereditary artinian rings with self dualities i in this paper we study the finitely generated indecomposable modules over an arbitrary extension over an artinian ring with self morita duality let a be an artinian ring with self duality and t an extension over a with kernel q see 11 chap xiv sect 21 such that qa and a q are isomorphic to injective hulls of top a and topla respectively such an a module q will be called quasi frobenius then it will be proved that

the space of embedded minimal surfaces of fixed genus in a 3 manifold ii multi valued graphs in disks this paper is the second in a series where we give a description of the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed 3 manifold the key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in r 3 we show here that if the curvature of such a disk becomes large at some point then it contains an almost flat multi valued graph nearby that continues almost all the way to the boundary this will be proved by showing the existence of small multi valued graphs near points of large curvature and then using the extension result for multi valued graphs proved in the first paper in this series there are two local models for embedded minimal disks by an embedded disk we mean a smooth injective map from the closed unit ball in r 2 into r 3 one model is the plane or more generally a minimal graph the other is a piece of a helicoid in the first four papers of this series we will show that every embedded minimal disk is either a graph of a function or is a double spiral staircase like a helicoid recall that a double spiral staircase consists of two spiral staircases that spiral together around a common axis one inside the other this will be done by showing that if the curvature is large at some point and hence the surface is not a graph then it is a double spiral staircase to prove that it is a double spiral staircase we will first prove that it is built out of n valued graphs where n is a fixed number these n valued graphs are like a single spiral staircase connecting n floors the existence of the n valued graphs was initiated in the first paper and will be completed here the third and fourth papers of this series will deal with how the multi valued graphs fit together and in particular prove regularity of the set of points of large curvature the axis of the double spiral staircase

on weakly bounded empirical processes let f be a class of functions on a probability space and let x 1 x k be independent random variables distributed according to we establish an upper bound that holds with high probability on rm sup f in f i f x i geq t for every t 0 and that depends on a natural geometric parameter associated with f we use this result to analyze the supremum of empirical processes of the form z f left k 1 sum i 1 k f p x i mathbb e f p right for p 1 using the geometry of f we also present some geometric applications of this approach based on properties of the random operator gamma k 1 2 sum i 1 k x i e i where x i i 1 k are sampled according to an isotropic log concave measure on mathbb r n

the space of embedded minimal surfaces of fixed genus in a 3 manifold i estimates off the axis for disks this paper is the first in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed riemannian 3 manifold the key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in r3 with the flat metric this study is undertaken here and completed in cm6 these local results are then applied in cm7 where we describe the general structure of fixed genus surfaces in 3 manifolds there are two local models for embedded minimal disks by an embedded disk we mean a smooth injective map from the closed unit ball in r2 into r3 one model is the plane or more generally a minimal graph the other is a piece of a helicoid in the first four papers of this series we will show that every embedded minimal disk is either a graph of a function or is a double spiral staircase where each staircase is a multi valued graph this will be done by showing that if the curvature is large at some point and hence the surface is not a graph then it is a double spiral staircase to prove that such a disk

on constructing stably equivalent functors in this case the left module bt becomes again a generalized tilting module with end t a 8 it should be noted here that the notion of generalized tilting modules is fairly general and the class of such modules contains all tilting modules of finite projective dimension in the sense of miyashita 6 and also all cotilting modules of finite injective dimension especially the injective cogenerator da is always a generalized tilting module even if the regular module a has infinite injective dimension on each side in the joint work with tachikawa 7 we proved the existence of a stably equivalent functor t a t b for any genuine tilting module defined by happel and ringle s by introducing torsion and torsion free resolutions of modules we have tried to generalize the existence

image super resolution via sparse representation this paper presents a new approach to single image superresolution based upon sparse signal representation research on image statistics suggests that image patches can be well represented as a sparse linear combination of elements from an appropriately chosen over complete dictionary inspired by this observation we seek a sparse representation for each patch of the low resolution input and then use the coefficients of this representation to generate the high resolution output theoretical results from compressed sensing suggest that under mild conditions the sparse representation can be correctly recovered from the downsampled signals by jointly training two dictionaries for the low and high resolution image patches we can enforce the similarity of sparse representations between the low resolution and high resolution image patch pair with respect to their own dictionaries therefore the sparse representation of a low resolution image patch can be applied with the high resolution image patch dictionary to generate a high resolution image patch the learned dictionary pair is a more compact representation of the patch pairs compared to previous approaches which simply sample a large amount of image patch pairs reducing the computational cost substantially the effectiveness of such a sparsity prior is demonstrated for both general image super resolution sr and the special case of face hallucination in both cases our algorithm generates high resolution images that are competitive or even superior in quality to images produced by other similar sr methods in addition the local sparse modeling of our approach is naturally robust to noise and therefore the proposed algorithm can handle sr with noisy inputs in a more unified framework

dictionary learning algorithms for sparse representation algorithms for data driven learning of domain specific overcomplete dictionaries are developed to obtain maximum likelihood and maximum a posteriori dictionary estimates based on the use of bayesian models with concave schur concave csc negative log priors such priors are appropriate for obtaining sparse representations of environmental signals within an appropriately chosen environmentally matched dictionary the elements of the dictionary can be interpreted as concepts features or words capable of succinct expression of events encountered in the environment the source of the measured signals this is a generalization of vector quantization in that one is interested in a description involving a few dictionary entries the proverbial 25 words or less but not necessarily as succinct as one entry to learn an environmentally adapted dictionary capable of concise expression of signals generated by the environment we develop algorithms that iterate between a representative set of sparse representations found by variants of focuss and an update of the dictionary using these sparse representations experiments were performed using synthetic data and natural images for complete dictionaries we demonstrate that our algorithms have improved performance over other independent component analysis ica methods measured in terms of signal to noise ratios of separated sources in the overcomplete case we show that the true underlying dictionary and sparse sources can be accurately recovered in tests with natural images learned overcomplete dictionaries are shown to have higher coding efficiency than complete dictionaries that is images encoded with an overcomplete dictionary have both higher compression fewer bits per pixel and higher accuracy lower mean square error

generalized tilting modules with finite injective dimension abstract let r be a left noetherian ring s a right noetherian ring and u r a generalized tilting module with s end u r the injective dimensions of u r and u s are identical provided both of them are finite under the assumption that the injective dimensions of u r and u s are finite we describe when the subcategory ext s n n u n is a finitely generated right s module is submodule closed as a consequence we obtain a negative answer to a question posed by auslander in 1969 finally some partial answers to wakamatsu tilting conjecture are given

local removable singularity theorems for minimal laminations in this paper we prove a local removable singularity theorem for certain minimal laminations with isolated singularities in a riemannian three manifold this removable singularity theorem is the key result used in our proof that a complete embedded minimal surface in r 3 with quadratic decay of curvature has nite total curvature