rational singularities and almost split sequences the main aim of this paper is to relate almost split sequences to singularity theory by showing that the mckay quiver built from the finite dimen sional representations of a finite subgroup g of gl 2 c where c is the complex numbers is isomorphic to the ar quiver of the reflexive modules of the quotient singularity associated with g over the past decade almost split sequences have been playing an increasingly important role in the representation theories of finite dimensional algebras and classical orders see 5 and 3 8 for basic existence theorems in these contexts while they have been known for some time to exist in higher dimensional situations 3 it has not been at all clear how they related to singularity theory if at all the main aim of this paper is to relate almost split sequences to singularity theory by showing that the mckay quiver built from the fimte dimensional complex represen tations of a finite subgroup g of gl 2 c where c is the complex numbers is isomorphic to the ar quiver of the reflexive modules over the quotient singularity r associated with g as in the case of finite dimensional algebras the ar quiver of reflexive r modules is defined in terms of the almost split sequences of reflexive r modules in the case g c sl 2 c mckay observed that the underlying graph of the mckay quiver with the trivial module removed is isomorphic to the desingulariza tion graph of the associated singularity various explanations of this phenomenon have been given by knorrer gonzalez sprinberg verdier 6 and artin verdier 1 which along the way have established most explicitly in 1 a natural one to one correspondence between the indecomposable reflexive r modules and the nodes of the desingularization graph but why the almost split sequences describe the edge of the desingularization graph still remains to be explained an effort has been made to make this paper as self contained as possible in particular no prior knowledge of almost split sequences is required before describing the contents of the six sections of this paper we fix some notation throughout this paper g is a finite group k an algebraically closed field of characteristic not dividing the order of g and v a two dimensional k representation of g setting s k x y the k algebra of formal power series the two dimen sional representation v gives a linear action of g on s as a group of k algebra automorphisms we denote by s g the skew group ring given by this action

on a semilinear elliptic neumann problem with asymmetric nonlinearities abstract we consider the neumann problem 1 1 below we extend therange of applicability of the sharp nonresonance condition derived in go om so as in particular allow asymmetric nonlinearities 1 introduction this paper is mainly concerned with the neumann problem i au g u h x inq 1 1 du du 0 on oil here a is the laplacian operator q is a bounded domain of rn n 1 with c1 1 boundary and unit exterior normal v g is a continuous function on r and h is taken in l f2 or in lp q in a previous work go om we showed that if g s 2g s 1 2 lim sup oo s s oo s or s oo then a necessary and sufficient condition for 1 1 to be solvable for any h l f2 is that the range of g be all of k here kx 0 k2 denote the distinct eigenvalues of a on q with neumann boundary condition andg s jo g t dt the surjectivity of g thus provides a nonresonance condi tion at the eigenvalue xx 0 we refer to go om for historical background

local superlinearity and sublinearity for the p laplacian we study the existence nonexistence and multiplicity of positive solutions for a family of problems updelta p u f lambda x u u in w 0 1 p upomega where is a bounded domain in mathbb r n n p and 0 is a parameter the family we consider includes the well known nonlinearities of ambrosetti brezis cerami type in a more general form namely lambda a x u q b x u r where 0 leqslant q p 1 r leqslant p 1 here the coefficient a x is assumed to be nonnegative but b x is allowed to change sign even in the critical case preliminary results of independent interest include the extension to the p laplacian context of the brezis nirenberg result on local minimization in w 0 1 p text and c 0 1 a c 1 alpha estimate for equations of the form updelta p u h x u with h of critical growth a strong comparison result for the p laplacian and a variational approach to the method of upper lower solutions for the p laplacian

periodic solutions of a second order ordinary differential equation a necessary and sufficient condition for nonresonance the nonlinearity g in 1 1 is a continuous function from r to r and the forcing term h is taken in l o 27r nonresonance means that 1 1 admits at least one solution x for any given h integrating eq 1 1 over a period one immediately sees that a necessary condition for nonresonance is that the function g be unbounded from above and from below on r it will appear later cf 1 6 that this unboundedness can be looked at as a condition relating the behaviour at infinity of the nonlinearity g with respect to the first eigenvalue 1 0 of the associated linear problem xf ax in 0 27r x 0 x 27r x 0 x1 271 1 2

scalable coding of depth maps with r d optimized embedding recent work on depth map compression has revealed the importance of incorporating a description of discontinuity boundary geometry into the compression scheme we propose a novel compression strategy for depth maps that incorporates geometry information while achieving the goals of scalability and embedded representation our scheme involves two separate image pyramid structures one for breakpoints and the other for sub band samples produced by a breakpoint adaptive transform breakpoints capture geometric attributes and are amenable to scalable coding we develop a rate distortion optimization framework for determining the presence and precision of breakpoints in the pyramid representation we employ a variation of the ebcot scheme to produce embedded bit streams for both the breakpoint and sub band data compared to jpeg 2000 our proposed scheme enables the same the scalability features while achieving substantially improved rate distortion performance at the higher bit rate range and comparable performance at the lower rates

efficient quadtree coding of images and video the quadtree data structure is commonly used in image coding to decompose an image into separate spatial regions to adaptively identify the type of quantizer used in various regions of an image the authors describe the theory needed to construct quadtree data structures that optimally allocate rate given a set of quantizers a lagrange multiplier method finds these optimal rate allocations with no monotonicity restrictions they use the theory to derive a new quadtree construction method that uses a stepwise search to find the overall optimal quadtree structure the search can be driven with either actual measured quantizer performance or ensemble average predicted performance they apply this theory to the design of a motion compensated interframe video coding system using a quadtree with vector quantization

positivity and lower and upper solutions for fourth order boundary value problems abstract this paper is devoted to the study the boundary value problem u 4 t f t u t for all t i 0 1 u 0 u 1 u 0 u 1 0 we prove the existence of at least one two or three solutions in the presence of a pair of not necessarily ordered lower and upper solutions the proof follows from maximum principles related to the operator u 4 m u and amann s three solutions theorem

existence and multiplicity of positive solutions for classes of singular elliptic pdes we consider the boundary value problem u g u u in u 0on where rn is a bounded domain is a nonnegative function in l such that 0 on some subset of of positive measure and g 0 r is continuous we establish the existence of three positive solutions when g 0 0 positone the graph of s 1g s is roughly s shaped and 0 we also prove that there exists at least one positive solution when g 0 0 and 0 1 we employ the method of sub super solutions to prove our results

nonresonance to the right of the first eigenvalue for the one dimensional p laplacian a quasilinear two point boundary value problem was considered this problem is given by the equation

lower and upper solutions for elliptic problems in nonsmooth domains abstract in this paper we prove some existence results of semilinear dirichlet problems in nonsmooth domains in presence of lower and upper solutions well ordered or not we first prove existence results in an abstract setting using degree theory we secondly apply them for domains with conical points

the three solutions theorem for p laplacian boundary value problems abstract the three solutions theorem is established for p laplacian boundary value problems in the presence of two pairs of lower and upper solutions which are well ordered by the relationship between the lower and upper solutions and degree theory in addition an application of the three solutions theorem is given

singular derived categories of factorial terminalizations and maximal modification algebras abstract let x be a gorenstein normal 3 fold satisfying elf with local rings which are at worst isolated hypersurface e g terminal singularities by using the singular derived category d sg x and its idempotent completion d sg x we give necessary and sufficient categorical conditions for x to be q factorial and complete locally q factorial respectively we then relate this information to maximal modification algebras mmas introduced in 20 by showing that if an algebra is derived equivalent to x as above then x is q factorial if and only if is an mma thus all rings derived equivalent to q factorial terminalizations in dimension three are mmas as an application we extend some of the algebraic results in 6 and 14 using geometric arguments

solutions for an operator equation under the conditions of pairs of paralleled lower and upper solutions abstract in this paper under the condition of two pairs of strict lower and upper solutions and using the concept of e 1 b limit increasing operator some multiplicity results for an operator equation are obtained by the method of the fixed point index

multiview image coding using depth layers and an optimized bit allocation in this paper we present a novel wavelet based compression algorithm for multiview images this method uses a layer based representation where the 3 d scene is approximated by a set of depth planes with their associated constant disparities the layers are extracted from a collection of images captured at multiple viewpoints and transformed using the 3 d discrete wavelet transform dwt the dwt consists of the 1 d disparity compensated dwt across the viewpoints and the 2 d shape adaptive dwt across the spatial dimensions finally the wavelet coefficients are quantized and entropy coded along with the layer contours to improve the rate distortion performance of the entire coding method we develop a bit allocation strategy for the distribution of the available bit budget between encoding the layer contours and the wavelet coefficients the achieved performance of our proposed scheme outperforms the state of the art codecs for several data sets of varying complexity

an optimal quadtree based motion estimation and motion compensated interpolation scheme for video compression we propose an optimal quadtree qt based motion estimator for video compression it is optimal in the sense that for a given bit budget for encoding the displacement vector field dvf and the qt segmentation the scheme finds a dvf and a qt segmentation which minimizes the energy of the resulting displaced frame difference dfd we find the optimal qt decomposition and the optimal dvf jointly using the lagrangian multiplier method and a multilevel dynamic program we introduce a new very fast convex search for the optimal lagrangian multiplier spl lambda which results in a very fast convergence of the lagrangian multiplier method the resulting dvf is spatially inhomogeneous since large blocks are used in areas with simple motion and small blocks in areas with complex motion we also propose a novel motion compensated interpolation scheme which uses the same mathematical tools developed for the qt based motion estimator one of the advantages of this scheme is the globally optimal control of the tradeoff between the interpolation error energy and the dvf smoothness another advantage is that no interpolation of the dvf is required since we directly estimate the dvf and the qt segmentation for the frame which needs to be interpolated we present results with the proposed qt based motion estimator which show that for the same dfd energy the proposed estimator uses about 25 fewer bits than the commonly used block matching algorithm we also experimentally compare the interpolated frames using the proposed motion compensated interpolation scheme with the reconstructed original frames

a constructive monotone iterative method for second order bvp in the presence of lower and upper solutions this paper concerns the monotone approximations of solutions of boundary value problems such as u f t u u 0 u 0 u 1 0 we consider linear iterative scheme in case f is lipschitz in u and satisfies a one sided lipschitz condition in u the initial approximations are lower and upper solutions which can be ordered one way a email protected we also consider the periodic and the dirichlet problems

monotone method for the neumann problem with lower and upper solutions in the reverse order this work is devoted to the study of the existence and approximation of solutions of the problem u f t u u u a u b 0 in the presence of lower and upper solutions in the reverse order to this end we prove anti maximum principles for the nonlinear operator u 2k u email protected

three dimensional flops and noncommutative rings limburgs univ ctr dept math b 3590 diepenbeek belgium van den bergh m limburgs univ ctr dept math univ campus b 3590 diepenbeek belgium vdbergh luc ac be

a positive operator approach to the neumann problem for a second order ordinary differential equation abstract in this paper the solvability of the second order neumann problem u t cu t f t u t 0 qquad u 0 a u r b is studied we suppose that there is a lower solution and an upper solution in the reversed order and we obtain optimal conditions infto assure the existence of a solution lying between and

flops and equivalences of derived categories for threefolds with only terminal gorenstein singularities the main purpose of this paper is to show that bridgeland s moduli space of perverse point sheaves for certain flopping contractions gives the flops and the fourier mukai transform given by the birational correspondence of the flop is an equivalence between bounded derived categories

unstable periodic solutions of a parabolic problem in the presence of non well ordered lower and upper solutions abstract we prove the solvability of the dirichlet periodic problem for a semilinear parabolic equation assuming the existence of a lower solution and an upper solution which do not satisfy the ordering condition our results yield the existence of unstable solutions localized by means of and requiring weak regularity of the coefficients of the equation

minimization of eigenvalues for a quasilinear elliptic neumann problem with indefinite weight we investigate the minimization of the positive principal eigenvalue of the problem pu m u p 2u in u 0 on over a class of sign changing weights m with m 0 it is proved that minimizers exist and satisfy a bang bang type property in dimension one we obtain a complete description of the minimizers this problem is motivated by applications from population dynamics

existence and localization of solution for second order elliptic bvp in presence of lower and upper solutions without any order abstract we prove an existence and localization result for a solution of a second order elliptic equations with dirichlet and regular oblique derivative boundary conditions the main result is obtained assuming the existence of a lower and an upper solution without particular order and a positively homogeneous asymptotic behaviour of the right hand member of the equation

arbitrarily shaped motion prediction for depth video compression using arithmetic edge coding depth image compression is important for compact representation of 3d visual data in texture plus depth format where texture and depth maps from one or more viewpoints are encoded and transmitted a decoder can then synthesize a freely chosen virtual view via depth image based rendering using nearby coded texture and depth maps as reference further depth information can be used in other image processing applications beyond view synthesis such as object identification segmentation and so on in this paper we leverage on the observation that neighboring pixels of similar depth have similar motion to efficiently encode depth video specifically we divide a depth block containing two zones of distinct values e g foreground and background into two arbitrarily shaped regions sub blocks along the dividing boundary before performing separate motion prediction mp while such arbitrarily shaped sub block mp can lead to very small prediction residuals resulting in few bits required for residual coding it incurs an overhead to transmit the dividing boundaries for sub block identification at decoder to minimize this overhead we first devise a scheme called arithmetic edge coding aec to efficiently code boundaries that divide blocks into sub blocks specifically we propose to incorporate the boundary geometrical correlation in an adaptive arithmetic coder in the form of a statistical model then we propose two optimization procedures to further improve the edge coding performance of aec for a given depth image the first procedure operates within a code block and allows lossy compression of the detected block boundary to lower the cost of aec with an option to augment boundary depth pixel values matching the new boundary given the augmented pixels do not adversely affect synthesized view distortion the second procedure operates across code blocks and systematically identifies blocks along an object contour that should be coded using sub block mp via a rate distortion optimized trellis experimental results show an average overall bitrate reduction of up to 33 over classical h 264 avc

monotone methods for periodic solutions of second order scalar functional differential equations an iterative method is presented which starting from a lower or from an upper periodic solution provides a monotone sequence converging to a periodic solution of 1 with some restrictions on the growth off the method extends to functional differential equations of type 1 two numerical examples with an a posteriori error analysis are given

fredholm alternative for the p laplacian in higher dimensions abstract in this paper we characterize the set of all right hand sides h c formula for which the boundary value problem p u 1 u p 2 u h in u 0 on has at least one weak solution u w01 p here 1 0 is the first eigenvalue of the p laplacian in particular we prove that for h 1 0 this problem is solvable and the energy functional e h u 1 p u p 1 p u p hu is unbounded from below

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

on dependent bit allocation for multiview image coding with depth image based rendering the encoding of both texture and depth maps of multiview images captured by a set of spatially correlated cameras is important for any 3 d visual communication system based on depth image based rendering dibr in this paper we address the problem of efficient bit allocation among texture and depth maps of multiview images more specifically suppose we are given a coding tool to encode texture and depth maps at the encoder and a view synthesis tool to construct intermediate views at the decoder using neighboring encoded texture and depth maps our goal is to determine how to best select captured views for encoding and distribute available bits among texture and depth maps of selected coded views such that the visual distortion of desired constructed views is minimized first in order to obtain at the encoder a low complexity estimate of the visual quality of a large number of desired synthesized views we derive a cubic distortion model based on basic dibr properties whose parameters are obtained using only a small number of viewpoint samples then we demonstrate that the optimal selection of coded views and quantization levels for corresponding texture and depth maps is equivalent to the shortest path in a specially constructed 3 d trellis finally we show that using the assumptions of monotonicity in the predictor s quantization level and distance suboptimal solutions can be efficiently pruned from the feasible space during solution search experiments show that our proposed efficient selection of coded views and quantization levels for corresponding texture and depth maps outperforms an alternative scheme using constant quantization levels for all maps commonly used in video standard implementations by up to 1 5 db moreover the complexity of our scheme can be reduced by at least 80 over the full solution search

higher dimensional auslander reiten theory on maximal orthogonal subcategories abstract we introduce the concept of maximal n 1 orthogonal subcategories over artin algebras and orders and develop n 1 dimensional auslander reiten theory on them we give the n auslander reiten translation and the n auslander reiten duality then show the existence of n almost split sequences and n fundamental sequences we give some examples

remarks on the lower and upper solutions method for second and third order periodic boundary value problems we study the solvability and the approximation of the solutions by monotone iteration of the second and third order periodic boundary value problems bvps u cu t u u 0 u email protected u 0 u email protected and email protected au bu t u u 0 u email protected u 0 u email protected u 0 u email protected in the presence of lower and upper solutions which may not satisfy the usual ordering condition to this end some maximum and antimaximum principles are stated and proved for certain linear differential operators naturally associated with the previously stated problems

positive solutions of a diffusive predator prey model with modified leslie gower and holling type ii schemes abstract in this paper we investigate the existence multiplicity and stability of positive solutions to a prey predator model with modified leslie gower and holling type ii schemes p u u a 1 b u c 1 v u k 1 in v v a 2 c 2 v u k 2 in u 0 v 0 in u v 0 on where r n n 1 is a bounded domain with a smooth boundary the parameters a i b c i k i i 1 2 are positive numbers u and v are the respective populations of prey and predator here we say u v with u v 0 is a positive solution of problem p if u v is a solution of p and u v 0 in

maximal modifications and auslander reiten duality for non isolated singularities we first generalize classical auslander reiten duality for isolated singularities to cover singularities with a one dimensional singular locus we then define the notion of ct modules for non isolated singularities and we show that these are intimately related to noncommutative crepant resolutions nccrs when r has isolated singularities ct modules recover the classical notion of cluster tilting modules but in general the two concepts differ then wanting to generalize the notion of nccrs to cover partial resolutions of operatorname spec r in the main body of this paper we introduce a theory of modifying and maximal modifying modules under mild assumptions all the corresponding endomorphism algebras of the maximal modifying modules for three dimensional gorenstein rings are shown to be derived equivalent we then develop a theory of mutation for modifying modules which is similar but different to mutations arising in cluster tilting theory our mutation works in arbitrary dimension and in dimension three the behavior of our mutation strongly depends on whether a certain factor algebra is artinian

wavelet based joint estimation and encoding of depth image based representations for free viewpoint rendering we propose a wavelet based codec for the static depth image based representation which allows viewers to freely choose the viewpoint the proposed codec jointly estimates and encodes the unknown depth map from multiple views using a novel rate distortion rd optimization scheme the rate constraint reduces the ambiguity of depth estimation by favoring piece wise smooth depth maps the optimization is efficiently solved by a novel dynamic programming along trees of integer wavelet coefficients the codec encodes the image and the depth map jointly to decrease their redundancy and to provide a rd optimized bitrate allocation between the two the codec also offers scalability both in resolution and in quality experiments on real data show the effectiveness of the proposed codec

positive solutions of elliptic problems with locally oscillating nonlinearities abstract we prove the existence of two unbounded sequences of strictly positive solutions obtained respectively as minimum and saddle points of the associated action functional of the elliptic problem u f x u in u 0 on assuming that an oscillatory behaviour at of s 2 0 s f x d occurs locally in