design of capacity approaching irregular low density parity check codes we design low density parity check ldpc codes that perform at rates extremely close to the shannon capacity the codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides our theoretical analysis of the codes is based on the work of richardson and urbanke see ibid vol 47 no 2 p 599 618 2000 assuming that the underlying communication channel is symmetric we prove that the probability densities at the message nodes of the graph possess a certain symmetry using this symmetry property we then show that under the assumption of no cycles the message densities always converge as the number of iterations tends to infinity furthermore we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution our codes are found by optimizing the degree structure of the underlying graphs we develop several strategies to perform this optimization we also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds

remarks on logarithmical regularity criteria for the navier stokes equations in this paper we establish some new logarithmically improved regularity criteria for the navier stokes equations in the homogeneous besov space b 1

multipliers paramultipliers and weak strong uniqueness for the navier stokes equations abstract in this article we describe spaces p such that if u is a weak in the sense of leray j leray sur le mouvement d un fluide visqueux remplissant l espace acta math 63 1934 193 248 solution of the navier stokes system for some initial data u 0 and if u belongs to p then u is unique in the class of weak solutions we say then that weak strong uniqueness holds it turns out that the proof of such results relies on the boundedness of a trilinear functional f l 2 h l 2 h p r where belong to 0 1 in order to find optimal conditions for the boundedness of f we are led to describing spaces of multipliers and of paramultipliers that is functions which map by classical pointwise product or by paraproduct a given sobolev spaces in another given sobolev space the study of these spaces enables us to give conditions for weak strong uniqueness which generalise all previously known results from the famous serrin criterion j serrin the initial value problem for the navier stokes equations in r e langer ed nonlinear problems univ of wisconsin press 1963 pp 69 98 to the recent conditions formulated by lemarie rieusset p g lemarie rieusset recent developments in the navier stokes problem chapman and hall 2003

reply to comment on invariants of differential equations defined by vector fields we undertook the determination of fundamental sets of invariants for differential equations defined by vector fields in a recent paper using the equivariant moving frame method a more exhaustive list for these invariants was found in a subsequent comment by francis valiquette we reply to that paper and point out some essential inconsistencies in the comment

weighted norm inequalities for pseudo differential operators with smooth symbols and their commutators abstract we establish weighted l p inequalities for pseudo differential operators with smooth symbols and their commutators by using a class of new weight functions which include the classical muckenhoupt weight functions our results improve essentially some well known results

multipliers between sobolev spaces and fractional differentiation abstract we characterize the pointwise multipliers which maps a sobolev space h r r d to a sobolev space h s r d in the case s r d 2

a remark on the regularity criterion for the 3d navier stokes equations involving the gradient of one velocity component abstract in this paper we consider the regularity criterion for weak solutions to the 3d navier stokes equations we show that if u 3 belongs to some multiplier spaces then the solution actually is smooth on 0 t in particular we have the regularity condition in the bmo spaces u 3 l 4 3 0 t bmo which improves previous results

logarithmically improved regularity criterion for the nematic liquid crystal flows in b 1 space abstract in this work we study the regularity criterion of the three dimensional nematic liquid crystal flows it is proved that if the vorticity satisfies 0 t t b 1 2 1 log e t b 1 d t where b 1 denotes the critical besov space then the solution u d becomes a regular solution on 0 t this result extends the recent regularity criterion obtained by fan and ozawa 2012 11

logarithmically improved regularity criterion for the harmonic heat flow and related equations abstract we use an interpolation inequality on besov spaces to show a logarithmically improved regularity criterion for the harmonic heat flow the landau lifshitz equations and the landau lifshitz maxwell system

the form boundedness criterion for the laplacian operator abstract in this paper we characterize the class of measurable functions or more generally real or complex valued distributions f such that the commutator operator c f f acting from h 1 r d to h 1 r d in other words we give a complete solution to the problem for the boundedness of the laplacian which is fundamental to quantum mechanics

efficient erasure correcting codes we introduce a simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs and analyze the algorithm by analyzing a corresponding discrete time random process as a result we obtain a simple criterion involving the fractions of nodes of different degrees on both sides of the graph which is necessary and sufficient for the decoding process to finish successfully with high probability by carefully designing these graphs we can construct for any given rate r and any given real number spl epsiv a family of linear codes of rate r which can be encoded in time proportional to ln 1 spl epsiv times their block length n furthermore a codeword can be recovered with high probability from a portion of its entries of length 1 spl epsiv rn or more the recovery algorithm also runs in time proportional to n ln 1 spl epsiv our algorithms have been implemented and work well in practice various implementation issues are discussed

on the regularity criterion of axisymmetric weak solutions to the 3d navier stokes equations abstract in this paper we consider the regularity criterion of axisymmetric weak solutions to the navier stokes equations in r 3 let u be an axisymmetric weak solution in r 3 0 t w curl u and w be the azimuthal component of w in the cylindrical coordinates it is proved that u becomes a regular solution if w l 2 2 s 0 t m 2 3 s where m 2 3 s is the critical morrey campanato space

extrinsic information transfer functions model and erasure channel properties extrinsic information transfer exit charts are a tool for predicting the convergence behavior of iterative processors for a variety of communication problems a model is introduced that applies to decoding problems including the iterative decoding of parallel concatenated turbo codes serially concatenated codes low density parity check ldpc codes and repeat accumulate ra codes exit functions are defined using the model and several properties of such functions are proved for erasure channels one property expresses the area under an exit function in terms of a conditional entropy a useful consequence of this result is that the design of capacity approaching codes reduces to a curve fitting problem for all the aforementioned codes a second property relates the exit function of a code to its helleseth klove levenshtein information functions and thereby to the support weights of its subcodes the relation is via a refinement of information functions called split information functions and via a refinement of support weights called split support weights split information functions are used to prove a third property that relates the exit function of a linear code to the exit function of its dual

parity check density versus performance of binary linear block codes new bounds and applications the moderate complexity of low density parity check ldpc codes under iterative decoding is attributed to the sparseness of their parity check matrices it is therefore of interest to consider how sparse parity check matrices of binary linear block codes can be a function of the gap between their achievable rates and the channel capacity this issue was addressed by sason and urbanke and it is revisited in this paper the remarkable performance of ldpc codes under practical and suboptimal decoding algorithms motivates the assessment of the inherent loss in performance which is attributed to the structure of the code or ensemble under maximum likelihood ml decoding and the additional loss which is imposed by the suboptimality of the decoder these issues are addressed by obtaining upper bounds on the achievable rates of binary linear block codes and lower bounds on the asymptotic density of their parity check matrices as a function of the gap between their achievable rates and the channel capacity these bounds are valid under ml decoding and hence they are valid for any suboptimal decoding algorithm the new bounds improve on previously reported results by burshtein and by sason and urbanke and they hold for the case where the transmission takes place over an arbitrary memoryless binary input output symmetric mbios channel the significance of these information theoretic bounds is in assessing the tradeoff between the asymptotic performance of ldpc codes and their decoding complexity per iteration under message passing decoding they are also helpful in studying the potential achievable rates of ensembles of ldpc codes under optimal decoding by comparing these thresholds with those calculated by the density evolution technique one obtains a measure for the asymptotic suboptimality of iterative decoding algorithms

symmetry and transformation properties of iterative ordinary differential equation abstract symmetries of linear iterative equations and new conditions on the infinitesimals are obtained regarding the expressions of the solutions in terms of the parameters of the source equation an ansatz is made on the original parameters we have also obtained an expression for the source parameters of the transformed equation under equivalence transformations we conducted this work with a special emphasis on second third and fourth order equations although some of our results are valid for equations of a general order

the kato problem for operators with weighted ellipticity we consider second order operators lw w 1 divawr with ellipticity controlled by a muckemphout a2 weight w we prove that the kato square root estimate l 1 2 w f l2 w kr fkl2 w holds in the weighted space l 2 w

moving coframes and symmetries of differential equations this paper studies the application of the structure theory of infinite dimensional pseudo groups to computing symmetries of differential equations the main tool is a combination of cartan s method of equivalence and the moving coframe method introduced by fels and olver our approach does not require a preliminary computation of infinitesimal defining systems their analysis and integration and uses differentiation and linear algebra operations only examples of its main features are given

a method for the equivalence group and its infinitesimal generators we give an effective method for the systematic determination of the equivalence group for any family of differential equations and for the determination of the infinitesimal generators of this group this is achieved by viewing the equivalence group as a projected subgroup related to the full symmetry group of the equation in which the coefficients specifying the family element in the equation are also considered as dependent variables the method is extended to provide the exact number of fundamental invariants of the equation without any prior calculation of these invariants it is then applied to review a number of results obtained in the recent literature on the subject of invariant functions of differential equations the method is also applied probably for the first time to a case of nonlinear equation

design of irregular ldpc codes with optimized performance complexity tradeoff the optimal performance complexity tradeoff for error correcting codes at rates strictly below the shannon limit is a central question in coding theory this paper proposes a numerical approach for the minimization of decoding complexity for long block length irregular low density parity check ldpc codes the proposed design methodology is applicable to any binary input memoryless symmetric channel and any iterative message passing decoding algorithm with a parallel update schedule a key feature of the proposed optimization method is a new complexity measure that incorporates both the number of operations required to carry out a single decoding iteration and the number of iterations required for convergence this paper shows that the proposed complexity measure can be accurately estimated from a density evolution and extrinsic information transfer chart analysis of the code a sufficient condition is presented for convexity of the complexity measure in the variable edge degree distribution when it is not satisfied numerical experiments nevertheless suggest that the local minimum is unique the results presented herein show that when the decoding complexity is constrained the complexity optimized codes significantly outperform threshold optimized codes at long block lengths within the ensemble of irregular codes

operators of harmonic analysis in weighted spaces with non standard growth last years there was increasing an interest to the so called function spaces with non standard growth known also as variable exponent lebesgue spaces for weighted such spaces on homogeneous spaces we develop a certain variant of rubio de francia s extrapolation theorem this extrapolation theorem is applied to obtain the boundedness in such spaces of various operators of harmonic analysis such as maximal and singular operators potential operators fourier multipliers dominants of partial sums of trigonometric fourier series and others in weighted lebesgue spaces with variable exponent there are also given their vector valued analogues

logarithmically improved regularity criteria for the navier stokes equations in lorentz spaces abstract in this paper we investigate three dimensional incompressible navier stokes equations we establish some logarithmically improved regularity criteria in term of the lorentz spaces to the navier stokes equations

efficient encoding of low density parity check codes low density parity check ldpc codes can be considered serious competitors to turbo codes in terms of performance and complexity and they are based on a similar philosophy constrained random code ensembles and iterative decoding algorithms we consider the encoding problem for ldpc codes more generally we consider the encoding problem for codes specified by sparse parity check matrices we show how to exploit the sparseness of the parity check matrix to obtain efficient encoders for the 3 6 regular ldpc code for example the complexity of encoding is essentially quadratic in the block length however we show that the associated coefficient can be made quite small so that encoding codes even of length n spl sime 100000 is still quite practical more importantly we show that optimized codes actually admit linear time encoding

achieving near capacity performance on multiple antenna channels with a simple concatenation scheme this paper proposes a capacity approaching yet simple scheme for multi input multiple output mimo channels the proposed scheme is based on a concatenation of a mixture of short memory length convolutional codes or repetition codes and a short and simple rate 1 linear block code followed by either 1 dimensional 1 d anti gray or gray mapping of quadrature phase shift keying qpsk modulation by interpreting the rate 1 code and the 1 d mapping as a multi d mapping performed over multiple transmit antennas the error performance is analyzed in two regions in the error floor region a tight union bound and the corresponding design criterion on the asymptotic performance are derived the bound provides a useful tool to predict the error performance at relatively low bit error rate ber values based on the obtained design criterion an optimal rate 1 code for each 1 d mapping is then constructed to achieve the best asymptotic performance in the turbo pinch off region by using extrinsic information transfer exit charts the most suitable mixed codes are selected for both symmetric and asymmetric antenna configurations it is demonstrated that the simple concatenation scheme can achieve a near capacity performance over the mimo channels furthermore its error performance is shown to be comparable to that obtained by using well designed irregular ldpc and ra codes and therefore the proposed scheme significantly outperforms a scheme employing a parallel concatenated turbo code simulation results in various cases are provided to verify the analysis

grassmannian packings from operator reed muller codes this paper introduces multidimensional generalizations of binary reed muller codes where the codewords are projection operators and the corresponding subspaces are widely separated with respect to the chordal distance on grassmannian space parameters of these grassmannian packings are derived and a low complexity decoding algorithm is developed by modifying standard decoding algorithms for binary reed muller codes the subspaces are associated with projection operators determined by pauli matrices appearing in the theory of quantum error correction and this connection with quantum stabilizer codes may be of independent interest the grassmannian packings constructed here find application in noncoherent wireless communication with multiple antennas where separation with respect to the chordal distance on grassmannian space guarantees closeness to the channel capacity it is shown that the capacity of the noncoherent multiple input multiple output mimo channel at both low and moderate signal to noise ratio snr under the constraint that only isotropically distributed unitary matrices are used for information transmission is closely approximated by these packings

capacity achieving ensembles for the binary erasure channel with bounded complexity we present two sequences of ensembles of nonsystematic irregular repeat accumulate ira codes which asymptotically as their block length tends to infinity achieve capacity on the binary erasure channel bec with bounded complexity per information bit this is in contrast to all previous constructions of capacity achieving sequences of ensembles whose complexity grows at least like the log of the inverse of the gap in rate to capacity the new bounded complexity result is achieved by puncturing bits and allowing in this way a sufficient number of state nodes in the tanner graph representing the codes we derive an information theoretic lower bound on the decoding complexity of randomly punctured codes on graphs the bound holds for every memoryless binary input output symmetric mbios channel and is refined for the binary erasure channel

singular invariant equation for the 1 1 fokker planck equation semi invariants for the linear parabolic equations with two independent variables time variable t and space variable x and one dependent variable u are derived under the transformation of the independent variables by using the infinitesimal method we also obtain the joint invariant equation for the above mentioned equation under equivalence transformation in fact we prove a necessary and sufficient condition for a 1 1 parabolic equation to be reducible via a local equivalence transformation to the one dimensional classical heat equation this result provides practical criteria for reduction finally examples of 1 1 fokker planck equations from applications are given to verify the results obtained

short non binary low density parity check codes for phase noise channels this paper considers the design of short non binary low density parity check ldpc codes over finite fields of order m for channels with phase noise in particular m ary differential phase shift keying dpsk modulated code symbols are transmitted over an additive white gaussian noise awgn channel with the wiener phase noise at the receiver side non coherent detection takes place with the help of a multi symbol detection algorithm followed by a non binary decoding step both the detector and the decoder operate on a joint factor graph as a benchmark finite length bounds and information rate expressions are computed and compared with the codeword error rate cer performance as well as the iterative threshold of the obtained codes as a result performance within 1 2 db from finite length bounds is obtained down to a cer of 10 3

local energy decay for the damped wave equation abstract we prove local energy decay for the damped wave equation on r d the problem which we consider is given by a long range metric perturbation of the euclidean laplacian with a short range absorption index under a geometric control assumption on the dissipation we obtain an almost optimal polynomial decay for the energy in suitable weighted spaces the proof relies on uniform estimates for the corresponding resolvent both for low and high frequencies these estimates are given by an improved dissipative version of mourre s commutators method

performance of weighted nonbinary repeat accumulate codes over gf q with q ary orthogonal modulation an approximate upper bound is derived to the maximum likelihood decoding threshold of weighted nonbinary repeat accumulate wnra codes over gf q with q ary orthogonal modulation and coherent detection over the additive white gaussian noise channel as with binary ra codes wnra codes show the potential for achieving the channel capacity with decreasing code rate for large enough q

characterization of the traces on the boundary of functions in magnetic sobolev spaces abstract we characterize the trace of magnetic sobolev spaces defined in a half space or in a smooth bounded domain in which the magnetic field a is differentiable and its exterior derivative corresponding to the magnetic field da is bounded in particular we prove that for d 1 and p 1 the trace of the magnetic sobolev space w a 1 p r d 1 is exactly w a 1 1 p p r d where a x a 1 a d x 0 for x r d with the convention a a 1 a d 1 when a c 1 r d 1 r d 1 we also characterize fractional magnetic sobolev spaces as interpolation spaces and give extension theorems from a half space to the entire space

computation of invariants of lie algebras by means of moving frames a new purely algebraic algorithm is presented for computation of invariants generalized casimir operators of lie algebras it uses the cartan method of moving frames and the knowledge of the group of inner automorphisms of each lie algebra the algorithm is applied in particular to computation of invariants of real low dimensional lie algebras a number of examples are calculated to illustrate its effectiveness and to make a comparison with the same cases in the literature bases of invariants of the real six dimensional solvable lie algebras with four dimensional nilradicals are newly calculated and listed in a table

design of repeat accumulate codes for iterative detection and decoding extrinsic information transfer exit charts are used to design systematic and nonsystematic repeat accumulate ra codes for iterative detection and decoding the convergence problems of nonsystematic ra codes are solved by introducing a biregular or doped layer of check nodes as examples such nonsystematic codes are designed for multi input multi output mimo fading channels and are shown to operate close to capacity

the equivalence problem for the euler bernoulli beam equation via cartan s method we completely solve the local point equivalence problem for the euler bernoulli beam equation using cartan s method of equivalence we obtain five equivalence classes for each equivalence class we establish the necessary and sufficient conditions for similarity and derive a basis of differential invariants as well as operators of invariant differentiation

finite length analysis of low density parity check codes on the binary erasure channel in this paper we are concerned with the finite length analysis of low density parity check ldpc codes when used over the binary erasure channel bec the main result is an expression for the exact average bit and block erasure probability for a given regular ensemble of ldpc codes when decoded iteratively we also give expressions for upper bounds on the average bit and block erasure probability for regular ldpc ensembles and the standard random ensemble under maximum likelihood ml decoding finally we present what we consider to be the most important open problems in this area

bounds on the number of iterations for turbo like ensembles over the binary erasure channel this paper provides simple lower bounds on the number of iterations which is required for successful message passing decoding of some important families of graph based code ensembles including low density parity check ldpc codes and variations of repeat accumulate codes the transmission of the code ensembles is assumed to take place over a binary erasure channel and the bounds refer to the asymptotic case where we let the block length tend to infinity the simplicity of the bounds derived in this paper stems from the fact that they are easily evaluated and are expressed in terms of some basic parameters of the ensemble which include the fraction of degree 2 variable nodes the target bit erasure probability and the gap between the channel capacity and the design rate of the ensemble this paper demonstrates that the number of iterations which is required for successful message passing decoding scales at least like the inverse of the gap in rate to capacity provided that the fraction of degree 2 variable nodes of these turbo like ensembles does not vanish hence the number of iterations becomes unbounded as the gap to capacity vanishes

generalized low density parity check codes based on hadamard constraints in this paper we consider the design and analysis of generalized low density parity check gldpc codes in awgn channels the gldpc codes are specified by a bipartite tanner graph as with standard ldpc codes but with the single parity check constraints replaced by general coding constraints in particular we consider imposing hadamard code constraints at the check nodes for a low rate approach termed ldpc hadamard codes we introduce a low complexity message passing based iterative soft input soft output siso decoding algorithm which employs the a posteriori probability app fast hadamard transform fht for decoding the hadamard check codes at each decoding iteration the achievable capacity with the gldpc codes is then discussed a modified ldpc hadamard code graph is also proposed we then optimize the ldpc hadamard code ensemble using a low complexity optimization method based on approximating the density evolution by a one dimensional dynamic system represented by an extrinsic mutual information transfer exit chart simulation results show that the optimized ldpc hadamard codes offer better performance in the low rate region than low rate turbo hadamard codes but also enjoy a fast convergence rate a rate 0 003 ldpc hadamard code with large block length can achieve a bit error rate ber performance of 10 5 at 1 44 db which is only 0 15 db away from the ultimate shannon limit 1 592 db and 0 24 db better than the best performing low rate turbo hadamard codes

the weighted lp boundedness of product type pseudodifferential operators abstract in this paper we consider non regular simple symbols and double symbols satisfying estimates of product type and give some sufficient conditions for the associated pseudodifferential operators to be bounded on weighted lp spaces where the weights considered here are the aq weights of product type in this paper we fill in the gap between the sufficient conditions for the unweighted lp boundedness and weighted lp boundedness some of the results are new even for the unweighted lp boundedness for symbols satisfying classical estimates

multipliers spaces muckenhoupt weights and pseudo differential operators abstract let x be a sufficiently regular function defined on r d r d the pseudo differential operator with symbol is defined on the schwartz class by the formula f f x r d x f e 2 i x d where f r d f x e 2 i x d x is the fourier transform of f in this paper we shall consider the regularity of the following type a x a 1 b x y x a y 1 n d where is suitable positive function and we prove boundedness results on multipliers spaces x r m h r l 2 for pseudo differential operators whose symbol x satisfies the regularity condition on x

iterative decoding of binary block and convolutional codes iterative decoding of two dimensional systematic convolutional codes has been termed turbo de coding using log likelihood algebra we show that any decoder can be used which accepts soft inputs including a priori values and delivers soft outputs that can be split into three terms the soft channel and a priori inputs and the extrinsic value the extrinsic value is used as an a priori value for the next iteration decoding algorithms in the log likelihood domain are given not only for convolutional codes but also for any linear binary systematic block code the iteration is controlled by a stop criterion derived from cross entropy which results in a minimal number of iterations optimal and suboptimal decoders with reduced complexity are presented simulation results show that very simple component codes are sufficient block codes are appropriate for high rates and convolutional codes for lower rates less than 2 3 any combination of block and convolutional component codes is possible several interleaving techniques are described at a bit error rate ber of 10 sup 4 the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple block convolutional component codes interleaver sizes less than 1000 and for three to six iterations

isospectral euler bernoulli beams with continuous density and rigidity functions isospectral systems are those that have exactly the same free vibration frequency spectrum with respect to a given boundary condition configuration in this paper seven different classes of inhomogeneous euler bernoulli beams with continuous density and flexural rigidity functions are found that are analytically solvable and that are isospectral with a homogeneous beam in the clamped clamped configuration of these exact solutions one class is isospectral with the homogeneous case in all ten distinct configurations obtainable from clamped sliding and two antiresonant end conditions whereas another class is isospectral in the six configurations with clamped hinged or free end conditions a connection with inverse problems is discussed

weighted sobolev spaces and pseudodifferential operators with smooth symbols absmacr let u i be the fefferman stein sharp function of u and for 1 0 such that the pointwise estimate au x cm u x holds for all x e ri and all schwartz functions u this estimate implies the boundedness of 0 order pseudodifferential operators on weighted lp spaces whenever the weight function belongs to muckenhoupt s class ap having established this we construct weighted sobolev spaces of fractional order in r and on a compact manifold prove a version of sobolev s theorem and exhibit coercive weighted estimates for elliptic pseudodifferential operators

dynamics of transversely vibrating beams using four engineering theories abstract in this paper the full development and analysis of four models for the transversely vibrating uniform beam are presented the four theories are the euler bernoulli rayleigh shear and timoshenko first a brief history of the development of each beam model is presented second the equation of motion for each model and the expressions for boundary conditions are obtained using hamilton s variational principle third the frequency equations are obtained for four sets of end conditions free free clamped clamped hinged hinged and clamped free the roots of the frequency equations are presented in terms of normalized wave numbers the normalized wave numbers for the other six sets of end conditions are obtained using the analysis of symmetric and antisymmetric modes fourth the orthogonality conditions of the eigenfunctions or mode shape and the procedure to obtain the forced response using the method of eigenfunction expansion is presented finally a numerical example is shown for a non slender beam to signify the differences among the four beam models

logarithmically improved regularity criteria for the navier stokes equations in multiplier spaces abstract the aim of this paper is to establish some logarithmically improved regularity criteria in term of the multiplier spaces to the navier stokes equations

bilinear forms on homogeneous sobolev spaces abstract in this paper we characterize the boundedness of the bilinear form defined by f g h s r h s r r s 2 f g x s 2 b x d x in the product of homogeneous sobolev spaces h s r h s r 0 s 1 2 we deduce a characterization of the space of pointwise multipliers from h s r to its dual h s r in terms of trace measures

a regularity criterion for the 3d magneto micropolar fluid equations in triebel lizorkin spaces abstract we consider the regularity criterion for the 3d magneto micropolar fluid equations in triebel lizorkin spaces it is proved that if u l p 0 t f q 2 q 3 0 with 2 p 3 q 2 3 2 q then the solution remains smooth in 0 t as a corollary we obtain the classical beal kato majda criterion that is the condition u l 1 0 t b 0 ensures the smoothness of the solution

invariants of differential equations defined by vector fields we determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold we also find all invariants and differential invariants for this group up to the second order a result on the characterization of classes of these equations by the invariant functions is also given

on partial regularity results for the navier stokes equations looking at the regularity results of scheffer respectively caffarelli kohn and nirenberg from a new point of view indicates that estimates for the pressure do not play an essential role in partial regularity results for the navier stokes equations

a note on regularity criterion for the navier stokes equations in terms of the pressure in this note a logarithmically improved regularity criterion for the navier stokes equations is established in terms of the pressure in the nonhomogeneous besov space

euler bernoulli beams from a symmetry standpoint characterization of equivalent equations abstract we completely solve the equivalence problem for euler bernoulli equation using lie symmetry analysis we show that the quotient of the symmetry lie algebra of the bernoulli equation by the infinite dimensional lie algebra spanned by solution symmetries is a representation of one of the following lie algebras 2 a 1 a 1 a 2 3 a 1 or a 3 3 a 1 each quotient symmetry lie algebra determines an equivalence class of euler bernoulli equations save for the generic case corresponding to arbitrary lineal mass density and flexural rigidity we characterize the elements of each class by giving a determined set of differential equations satisfied by physical parameters lineal mass density and flexural rigidity for each class we provide a simple representative and we explicitly construct transformations that maps a class member to its representative the maximally symmetric class described by the four dimensional quotient symmetry lie algebra a 3 3 a 1 corresponds to euler bernoulli equations homeomorphic to the uniform one constant lineal mass density and flexural rigidity we rigorously derive some non trivial and non uniform euler bernoulli equations reducible to the uniform unit beam our models extend and emphasize the symmetry flavor of gottlieb s iso spectral beams h p w gottlieb isospectral euler bernoulli beam with continuous density and rigidity functions proc r soc lond ser a math phys eng sci 413 1987 235 250

regularity criteria in terms of pressure for the 3 d navier stokes equations in a generic domain we consider the 3 d navier stokes equations in the half space 3 or a bounded domain with smooth boundary or else an exterior domain with smooth boundary some new sufficient conditions on pressure or the gradient of pressure for the regularity of weak solutions to the navier stokes equations are obtained

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

note on prodi serrin ladyzhenskaya type regularity criteria for the navier stokes equations in this article we prove new regularity criteria of the prodi serrin ladyzhenskaya type for the cauchy problem of the three dimensional incompressible navier stokes equations our results improve the classical lr 0 t ls regularity criteria for both velocity and pressure by factors of certain negative powers of the scaling invariant norms u l 3 and u h 1 2

weak and strong type estimates for the multilinear pseudo differential operators abstract in this paper we investigate the boundedness of the multilinear pseudo differential operator t first we establish the local exponential decay estimates for t in terms of the corresponding commutators t b we obtain the local subexponential decay estimates secondly we derive the weighted mixed weak type inequality for t which parallels sawyer s conjecture for calderon zygmund operators and covers the endpoint weighted inequalities last but not least we present the sharp weighted estimates for t and t b it is worth mentioning that our results are totally new even in the linear case

generalized navier stokes equations with initial data in local q type spaces abstract in this paper we establish a link between leray mollified solutions of the three dimensional generalized navier stokes equations and mild solutions for initial data in the adherence of the test functions for the norm of q loc 1 r 3 this result applies to the usual incompressible navier stokes equations

design of low density parity check codes for modulation and detection a coding and modulation technique is studied where the coded bits of an irregular low density parity check ldpc code are passed directly to a modulator at the receiver the variable nodes of the ldpc decoder graph are connected to detector nodes and iterative decoding is accomplished by viewing the variable and detector nodes as one decoder the code is optimized by performing a curve fitting on extrinsic information transfer charts design examples are given for additive white gaussian noise channels as well as multiple input multiple output mimo fading channels where the receiver but not the transmitter knows the channel for the mimo channels the technique operates within 1 25 db of capacity for various antenna configurations and thereby outperforms a scheme employing a parallel concatenated turbo code by wide margins when there are more transmit than receive antennas

a logarithmic improvement of regularity criterion for the mhd equations in terms of the pressure abstract in this paper we prove a logarithmic improvement of regularity criterion only in terms of the pressure in morrey campanato spaces for the cauchy problem to the incompressible mhd equations

invariants of a family of third order ordinary differential equations we consider a family of third order ordinary differential equations odes with the cubic nonlinearity in second order derivative which is closed with respect to contact transformations using lie s infinitesimal method we construct the basis of differential invariants for this family of equations and the operators of invariant differentiation invariants provide a simple way of finding the equations which may be equivalent to a given nonlinear ode and the transformation connecting two equivalent equations some examples are given to illustrate the results obtained in the paper

a nonlinear plancherel theorem with applications to global well posedness for the defocusing davey stewartson equation and to the inverse boundary value problem of calderon author s nachman adrian i regev idan tataru daniel i abstract we prove a plancherel theorem for a nonlinear fourier transform in two dimensions arising in the inverse scattering method for the defocusing davey stewartson ii equation we then use it to prove global well posedness and scattering in l 2 for defocusing dsii this plancherel theorem also implies global uniqueness in the inverse boundary value problem of calder x27 on in dimension 2 for conductivities sigmag0 with log sigma in dot h 1 the proof of the nonlinear plancherel theorem includes new estimates on classical fractional integrals as well as a new result on l 2 boundedness of pseudo differential operators with non smooth symbols valid in all dimensions

a class of solvable lie algebras and their casimir invariants a nilpotent lie algebra with an n 1 dimensional abelian ideal is studied all indecomposable solvable lie algebras with as their nilradical are obtained their dimension is at most n 2 the generalized casimir invariants of and of its solvable extensions are calculated for n 4 these algebras figure in the petrov classification of einstein spaces for larger values of n they can be used in a more general classification of riemannian manifolds

weighted norm inequalities for pseudo pseudodifferential operators defined by amplitudes abstract we prove weighted norm inequalities for pseudodifferential operators with amplitudes which are only measurable in the spatial variables the result is sharp even for smooth amplitudes nevertheless in the case when the amplitude contains the oscillatory factor e i 1 the result can be substantially improved we extend the l p boundedness of pseudo pseudodifferential operators to certain weights end point results are obtained when the amplitude is either smooth or satisfies a homogeneity condition in the frequency variable our weighted norm inequalities also yield the boundedness of commutators of these pseudodifferential operators with functions of bounded mean oscillation