observations of measured hydraulic conductivity in two artificial confined aquifers with boundaries recent theoretical efforts have provided strong evidence that the effective conductivity varies with distance between monitoring wells and a point source in the present study two artificial confined aquifers were subjected to steady state pumping tests analysis of these experimental results indicates dependence of the estimated hydraulic conductivity on distance from the pumping well and the form of the physical heterogeneity present as the precision of the measured hydraulic heads within the experimental apparatus was limited to 0 5 mm the experimental results were reproduced numerically although some minor discrepancies were noted between the numerical and experimental results the primary observations were consistent in all cases the thiem equation was used in combination with image well theory to account for the boundary conditions to analyze the response at 25 observation points 87 observation points used in the numerical experiments providing a number of estimates of the hydraulic conductivity within each experiment it was observed that the variance of the conductivity estimate decreased with the distance between the two observation wells used in the calculation further the mean conductivity varied with distance from the pumping well in an experiment in which the medium was constructed to mimic a second order stationary random field the mean conductivity approached at large distance from the pumping well the conductivity for a random anisotropic medium under mean uniform flow in an experiment in which the medium was constructed to mimic a structured scaled medium the mean conductivity decreased at large distance from the pumping well below the geometric mean and approached a value between the geometric and the harmonic means of the conductivity field while the boundary conditions make it difficult to fully analyze the large dimension behavior of these two media it is clear from the results that the structured medium performs quite differently than the random field at large measurement scale

improved conditions for single point blow up in reaction diffusion systems abstract we study positive blowing up solutions of the system u t u v p v t v u q as well as of some more general systems for any p q 1 we prove single point blow up for any radially decreasing positive and classical solution in a ball this improves on previously known results in 3 directions i no type i blow up assumption is made and it is known that this property may fail ii no equidiffusivity is assumed i e any 0 is allowed iii a large class of nonlinearities f u v g u v can be handled which need not follow a precise power behavior as a side result we also obtain lower pointwise estimates for the final blow up profiles

some new classes of quantum mds codes from constacyclic codes quantum maximum distance separable mds codes form an important family of quantum codes in this paper using hermitian construction and classical constacyclic codes we construct six classes of quantum mds codes two of these six classes of quantum mds codes have larger minimum distance than the ones available in the literature most of these quantum mds codes are new in the sense that their parameters are not covered by the codes available in the literature

nonbinary quantum codes we present several results on quantum codes over general alphabets that is in which the fundamental units may have more than two states in particular we consider codes derived from finite symplectic geometry assumed to have additional global symmetries from this standpoint the analogs of calderbank shor steane codes and of gf 4 linear codes turn out to be special cases of the same construction this allows us to construct families of quantum codes from certain codes over number fields in particular we get analogs of quadratic residue codes including a single error correcting code encoding one letter in five for any alphabet size we also consider the problem of fault tolerant computation through such codes generalizing ideas of gottesman see phys rev a vol 57 no 1 p127 37 1998

complete blow up after tmax for the solution of a semilinear heat equation soit un sous ensemble ouvert borne de r n a frontiere lisse on considere le probleme u t u f u dans o t u 0 sur o t u x o u 0 x x ou f r r est localement de lipschitz non decroissante et f o 0

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

nonbinary stabilizer codes over finite fields one formidable difficulty in quantum communication and computation is to protect information carrying quantum states against undesired interactions with the environment to address this difficulty many good quantum error correcting codes have been derived as binary stabilizer codes fault tolerant quantum computation prompted the study of nonbinary quantum codes but the theory of such codes is not as advanced as that of binary quantum codes this paper describes the basic theory of stabilizer codes over finite fields the relation between stabilizer codes and general quantum codes is clarified by introducing a galois theory for these objects a characterization of nonbinary stabilizer codes over fq in terms of classical codes over fq 2 is provided that generalizes the well known notion of additive codes over f4 of the binary case this paper also derives lower and upper bounds on the minimum distance of stabilizer codes gives several code constructions and derives numerous families of stabilizer codes including quantum hamming codes quadratic residue codes quantum melas codes quantum bose chaudhuri hocquenghem bch codes and quantum character codes the puncturing theory by rains is generalized to additive codes that are not necessarily pure bounds on the maximal length of maximum distance separable stabilizer codes are given a discussion of open problems concludes this paper

quantum reed muller codes a set of quantum error correcting codes based on classical reed muller codes is described the codes have parameters n k d 2 sup r s sup r c r t 2 spl sigma sub i 0 sup t 1 c r i 2 sup t 2 sup t 1

new quantum mds codes from negacyclic codes let q be an odd prime power based on classical negacyclic codes we construct two classes of quantum maximum distance separable mds codes with parameters q2 1 q2 2d 3 d q where q 1 mod 4 and 2 d q 1 is even and q2 1 2 q2 1 2 2d 2 d q where 3 d q is odd some of these quantum mds codes are new in the sense that their parameters are different from all the previously known ones

blow up of solutions of nonlinear heat equations on considere des solutions de u t u xx f u pour a 0 u a t 0 pour t 0 u x 0 x pour a x a ou f u est une fonction superlineaire continument differentiable

nonbinary quantum stabilizer codes we define and show how to construct nonbinary quantum stabilizer codes our approach is based on nonbinary error bases it generalizes the relationship between self orthogonal codes over f sub 4 and binary quantum codes to one between self orthogonal codes over f q sup 2 and q ary quantum codes for any prime power q

enlargement of calderbank shor steane quantum codes it is shown that a classical error correcting code c n k d which contains its dual c sup spl perp spl sube c and which can be enlarged to c n k k 1 d can be converted into a quantum code of parameters n k k n min d 3d 2 this is a generalization of a previous construction it enables many new codes of good efficiency to be discovered examples based on classical bose chaudhuri hocquenghem bch codes are discussed

nonlocal properties of nonuniform averaged flows in heterogeneous media the properties of nonuniform average potential flows in media of stationary random conductivity are studied the mathematical model of average flow is derived as a system of governing equations to be satisfied by mean velocity and mean head the averaged darcy s law determines the effective conductivity as an integral operator of the convolution type relating the mean velocity to the mean head gradient in a nonlocal way in fourier domain the mean velocity is proportional to the mean head gradient the coefficient of proportionality is referred to as the effective conductivity tensor and is derived by perturbation methods in terms of functionals of the conductivity spatial moments it is shown that the effective conductivity cannot be defined uniquely for potential flows however this nonuniqueness does not affect the spatial distribution of the mean head and velocity analytical expressions of the effective conductivity tensor are derived for two and three dimensional flows and for exponential and gaussian correlations of isotropic conductivity the fundamental solution of the governing equations in effective media mean green function is calculated for the same cases two new asymptotic models of the averaged darcy s law are developed to be applicable to large and small scales of heterogeneity several asymptotic expansions of tue two dimensional and three dimensional mean green functions are derived for exponential and gaussian correlations

effective groundwater model parameter values influence of spatial variability of hydraulic conductivity leakance and recharge a stochastic simulation approach was used to inspect the influence of spatial variability in aquifer and recharge properties on effective averaged groundwater model parameters a two dimensional unconfined aquifer model was set up for an area similar to that near livermore california monte carlo simulations were generated for different sets of spatial correlation structures assuming stationarity and an exponential semivariogram for each monte carlo realization groundwater flow was simulated and estimates of the means and standard deviations of the hydraulic head were obtained nine different cases were studied involving different correlation lengths of hydraulic conductivity conditional simulations parameter zonation and spatial recharge distributions results indicate that there is no single uniform value of the hydraulic conductivity that reproduces the expected head values spatial averaging to obtain effective values was based on the p norm power norm which ranges from 1 for the arithmetic mean to 1 for the harmonic mean with 0 representing the geometric mean for the unconditional case the 0 4 p norm seems to best reproduce the expected values the 0 2 p norm gave the best result for the conditional simulation case if the geometric mean were used instead the heads would deviate from the expected heads by about 2 m near the wells some value between the arithmetic and the geometric means of riverbed leakance will give results close to the expected head values the arithmetic mean of aerially distributed recharge produced results with small deviations from the expected head values these particular results may only apply to this modeled system as the pumping pattern and magnitudes exhibited strong influences on zones of maximum head deviation the implication of this work is that if an effective hydraulic conductivity is used in a simulation model it must be selected for a particular set of wells and pumping rates in an aquifer with significant pumping centers the best effective value for a two dimensional unconfined flow model is most likely between the geometric and harmonic mean if the wells are turned off the effective hydraulic conductivity for that flow model reverts to the geometric mean

pseudo cyclic codes and the construction of quantum mds codes constacyclic codes which generalize the classical cyclic codes have played important roles in recent constructions of many new quantum maximum distance separable mds codes however the mathematical mechanism may not have been fully understood in this paper we use pseudo cyclic codes which is a further generalization of constacyclic codes to construct the quantum mds codes we can not only provide a unified explanation of many previous constructions but also produce some new quantum mds codes

application of classical hermitian self orthogonal mds codes to quantum mds codes in this paper we first construct several classes of classical hermitian self orthogonal maximum distance separable mds codes through these classical codes we are able to obtain various quantum mds codes it turns out that many of our quantum codes are new in the sense that the parameters of our quantum codes cannot be obtained from all previous constructions

a new construction of quantum error correcting codes in this paper we present a characterization of binary and non binary quantum error correcting codes based on this characterization we introduce a method to construct p ary quantum codes using boolean functions satisfying a system of certain quadratic relations as a consequence of the construction we are able to construct quantum codes of minimum distance 2 in particular we produce a class of binary quantum n 2 n 2 1 2 n 1 n 1 2 2 codes for odd length 5 for n 11 this improves the result by rains in quantum codes of minimal distance two 1999 showing the existence of binary quantum n 3 2 n 4 2 codes for odd n 5 moreover our binary quantum n 2 n 2 1 2 n 1 n 1 2 2 codes of odd length achieve the singleton bound asymptotically finally based on our characterization some propagation rules of quantum codes are proposed and the rules are similar to those in classical coding theory it turns out that some new quantum codes are found through these propagation rules

numbers of zeros on invariant manifolds in reaction diffusion equations on considere l equation a une dimension u t u xx f x u t 0 0 x 1 avec les conditions aux limites de dirichlet u t 0 t 1 0 ou f 0 1 r r est bc 1 c k k 1

blowup solutions for a reaction diffusion system with exponential nonlinearities abstract we consider the following parabolic system whose nonlinearity has no gradient structure t u u e p v t v v e q u u 0 u 0 v 0 v 0 p q 0 in the whole space r n we show the existence of a stable blowup solution and obtain a complete description of its singularity formation the construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude in particular our analysis uses neither the maximum principle nor the classical methods based on energy type estimates which are not supported in this system the stability is a consequence of the existence proof through a geometrical interpretation of the quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem

blow up set of type i blowing up solutions for nonlinear parabolic systems we consider the blow up problem for systems of nonlinear parabolic inequalities and establish a criterion for the location of the blow up set our criterion enables us to obtain sufficient conditions for the boundedness of the blow up set and no boundary blow up for type i blowing up solutions

asymptotic bounds on quantum codes from algebraic geometry codes we generalize a characterization of p ary p is a prime quantum codes given by feng and xing to q ary q is a prime power quantum codes this characterization makes it possible to convert an asymptotic bound of stichtenoth and xing for nonlinear algebraic geometry codes to a quantum asymptotic bound besides we also investigate the asymptotic behavior of quantum codes

constacyclic symbol pair codes lower bounds and optimal constructions symbol pair codes introduced by cassuto and blaum 2010 are designed to protect against pair errors in symbol pair read channels the higher the minimum pair distance the more pair errors the code can correct maximum distance separable mds symbol pair codes are optimal in the sense that pair distance cannot be improved for given length and code size the contribution of this paper is twofold first we present three lower bounds for the minimum pair distance of constacyclic codes the first two of which generalize the previously known results due to cassuto and blaum 2011 and kai et al 2015 the third one exhibits a lower bound for the minimum pair distance of repeated root cyclic codes second we obtain new mds symbol pair codes with minimum pair distance seven and eight through repeated root cyclic codes

a construction of new mds symbol pair codes recently symbol pair codes are proposed to protect against pair errors in symbol pair read channels one main task in symbol pair coding theory is to design codes with large minimum pair distance maximum distance separable mds symbol pair codes are optimal in the sense they attain maximal minimum pair distance in this paper based on constacyclic codes we construct some new mds symbol pair codes with minimum pair distance five and six compared with classical q ary mds codes the constructed mds symbol pair codes have length up to q 2 q 1

on binary constructions of quantum codes recently steane 1998 proposed an enlargement of the calderbank shor steane construction here we further improve on the parameters obtained from his construction present examples of new codes and analyze asymptotical non constructive bounds

convergence asymptotic periodicity and finite point blow up in one dimensional semilinear heat equations abstract we consider the initial value problem for the semilinear heat equation u t u xx f u t 0 x l t 0 under the dirichlet the neumann or the periodic boundary conditions we show that each solution whether it exists globally for t 0 or blows up in a finite time possesses an asymptotic profile in a certain sense and tends to this profile as time increases in the special case where f u t t f u t for some t 0 among other things the above statement is interpreted as saying that any bounded global solution converges as t to a time t periodic solution having some specific spatial structure in the case where the solution blows up in a finite time say at t t 0 assuming simply that f is a smooth function satisfying some growth conditions and that the initial data is a nonconstant bounded function we prove that the blow up set is a finite set and that lim t t 0 u x t x exists with being a smooth function having at most finitely many singular points

the dynamics of rotating waves in scalar reaction diffusion equations on etudie l attracteur compact maximal pour l equation u t u xx f u u x avec des conditions aux limites periodiques on montre qu un ensemble limite contient une onde de rotation

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

analytic solutions of some nonlinear diffusion equations on considere u t u u 2 x n u 0 avec des fonctions initiales non constantes est un domaine borne de r n a frontiere lisse

constacyclic codes and some new quantum mds codes one central theme in quantum error correction is to construct quantum codes that have a large minimum distance quantum maximal distance separable mds codes are optimal in the sense they attain maximal minimum distance recently constructing quantum mds codes has received much attention and seems to become more and more difficult in this paper based on classical constacyclic codes we construct some new quantum mds codes by employing the hermitian construction compared with the known quantum mds codes these quantum mds codes have much larger minimum distance

on quantum and classical bch codes classical bose chaudhuri hocquenghem bch codes that contain their euclidean or hermitian dual codes can be used to construct quantum stabilizer codes this correspondence studies the properties of such codes it is shown that a bch code of length n can contain its dual code only if its designed distance delta o radicn and the converse is proved in the case of narrow sense codes furthermore the dimension of narrow sense bch codes with small design distance is completely determined and consequently the bounds on their minimum distance are improved these results make it possible to determine the parameters of quantum bch codes in terms of their design parameters

prediction of steady state flow in nonuniform geologic media by conditional moments exact nonlocal formalism effective conductivities and weak approximation we consider the effect of measuring randomly varying local hydraulic conductivities k x on one s ability to predict steady state flow within a bounded domain driven by random source and boundary functions more precisely we consider the prediction of local hydraulic head h x and darcy flux q x by means of their unbiased ensemble moments h x and q x conditioned on measurements of k x these predictors satisfy a deterministic flow equation in which q x x h x r x where x is a relatively smooth unbiased estimate of k x and r x is a residual flux we derive a compact integral expression for r x which is rigorously valid for a broad class of k x fields including fractals it demonstrates that q x is nonlocal and non darcian so that an effective hydraulic conductivity does not generally exist we show analytically that under uniform mean flow the effective conductivity may be a scalar a symmetric or a nonsymmetric tensor or a set of directional scalars which do not form a tensor we demonstrate numerically that in two dimensional mean radial flow it may increase from the harmonic mean of k x near interior and boundary sources to the geometric mean far from such sources for cases where r x can neither be expressed nor approximated by a local expression we propose a weak integral approximation closure which appears to work well in media with pronounced heterogeneity and improves as the quantity and quality of k x measurements increase the nonlocal deterministic flow equation can be solved numerically by standard methods our theory shows clearly how the scale of grid discretization should relate to the scale quantity and quality of available data after providing explicit approximations for the second moments of head and flux prediction errors we conclude by discussing practical methods to compute x from noisy measurements of k x and to calculate required second moments of the associated estimation errors when k x is lognormal

spatial averaging of transmissivity in heterogeneous fields with flow toward a well steady state nonuniform flow in a heterogeneous medium is investigated using a combined numerical empirical approach the objective of the study is to determine the effective transmissivity of a square field with a constant head external boundary and a constant rate well at its center transmissivity at the point scale is modeled as an isotropic and multivariate lognormal spatial random function block transmissivities are defined using an empirical scaling up of point support values within the field the scaling process is a weighted spatial geometric averaging where log transmissivities are weighted by the inverse square of their distance from the well block transmissivities thus obtained and true effective values calculated using a numerical flow model are compared for realizations a finely discretized transmissivity field the principal finding of this study is that effective transmissivities obtained from the numerical flow model and block values obtained by spatial averaging are in excellent agreement for low to moderate variances of log transmissivity and moderately eccentric field geometries the geostatistical model for point scale transmissivity and the deterministic spatial averaging law are combined in order to create a geostatistical model for transmissivity at the block scale exact expressions are derived for the ensemble moments of block averaged transmissivities and well drawdowns in both the nonconditional case and the conditional case when transmissivity at the well bore is known the ensemble mean of block averaged transmissivity is found to decrease from the ensemble arithmetic mean toward the ensemble geometric mean as the field size becomes large compared to the integral range of spatial correlation this result contradicts earlier published work numerical results presented in this study also disagree with the analytical results of a recent stochastic analysis of specific discharge in radial systems

quantum codes from concatenated algebraic geometric codes we apply steane s enlargement of the calderbank shor steane css codes and additive codes over f sub 4 to concatenated algebraic geometric codes to construct many good quantum codes with fewer restrictions on the parameters compared to some known quantum codes some of the quantum codes we have constructed are either optimal or have parameters as good as the best known codes while some have parameters better than those obtained from other known constructions

generalization of steane s enlargement construction of quantum codes and applications we generalize steane s enlargement construction of binary quantum codes to q ary quantum codes we then apply this result to bch codes and the study of asymptotic bounds and obtain improvements to the quantum bch codes constructed by aly and klappenecker and the quantum asymptotic bounds from algebraic geometry codes obtained by feng ling and xing

generation of random fields characterized by discrete regions of constant value a new simulator is introduced which provides realizations of parameter fields characterized by randomly sized regions within which the parameter value is constant this generator produces second order stationary random fields with specified mean and correlation structure the structure of the simulated field is significantly different than that which can be obtained by previously existing techniques this new structure provides a valuable simulation tool for discretely varying fields such as required when modeling complex lithologies one application of this method utilizing ovals in two dimensions and ellipsoids in three dimensions is demonstrated for grids in which a finite number of simulation points are to be generated this technique is compared with a recently published turning bands routine and is faster in terms of cpu time for large values of i l where i is the integral scale of the correlation structure and l is the length of the longest side of the simulation grid for small values of i l the turning bands routine is faster conditional simulation based on kriging has been studied for this method the resulting random field varies continuously rather than discretely a second method of conditioning which produces discretely varying random fields is suggested but not examined computationally

single point blow up for parabolic systems with exponential nonlinearities and unequal diffusivities abstract we study positive blowing up solutions of systems of the form u t 1 u e p v v t 2 v e q u with 1 2 0 and p q 0 we prove single point blow up for large classes of radially decreasing solutions this answers a question left open in a paper of friedman and giga 1987 where the result was obtained only for the equidiffusive case 1 2 and the proof depended crucially on this assumption

application of constacyclic codes to quantum mds codes quantum maximum distance separable mds codes form an important class of quantum codes to get q ary quantum mds codes one of the effective ways is to find linear mds codes c over f q2 satisfying c h c where c h denotes the hermitian dual code of c for a linear code c of length n over f q2 we say that c is a dual containing code if c h c and c f q2 n several classes of new quantum mds codes with relatively large minimum distance have been produced through dual containing constacyclic mds codes these works motivate us to make a careful study on the existence conditions for dual containing constacyclic codes we obtain necessary and sufficient conditions for the existence of dual containing constacyclic codes four classes of dual containing constacyclic mds codes are constructed and their parameters are computed consequently the quantum mds codes are derived from these parameters the quantum mds codes exhibited here have minimum distance bigger than the ones available in the literature

some families of asymmetric quantum codes and quantum convolutional codes from constacyclic codes abstract quantum maximal distance separable mds codes are an important class of quantum codes recently many scholars utilize constacyclic codes to construct some quantum mds codes in this paper several new families of optimal asymmetric quantum codes and optimal quantum convolutional codes are constructed by using constacyclic codes moreover these quantum codes constructed in this paper are different from the ones in the literature

quantum codes 6 2 3 sub p and 7 3 3 sub p p spl ges 3 exist we prove that nonbinary quantum stabilizer codes with parameters n k d sub p 6 2 3 sub p and 7 3 3 sub p exist for all odd primes p by using graph machinery given by schlingemann and werner see phys rev a vol 65 no 012308 2001 with a little number theory and combinatorics

pseudocyclic maximum distance separable codes the n k pseudocyclic maximum distance separable mds codes modulo x sup n a over gf q are considered suppose that n is a divisor of q 1 if n is odd pseudocyclic mds codes exist for all k however if n is even nontrivial pseudocyclic mds codes exist for odd k but not for even k if a is a quadratic residue in gf q and they exist for even k but not for odd k if a is not a quadratic residue in gf q also considered is the case when n is a divisor of q 1 and it is shown that pseudocyclic mds codes exist if and only if the multiplicative order of a divides q 1 n and that when this condition is satisfied such codes exist for all k if the condition is not satisfied every pseudocyclic code of length n is the result of interleaving a shorter pseudocyclic code

on nonexistence of type ii blowup for a supercritical nonlinear heat equation in this paper we study blowup of radially symmetric solutions of the nonlinear heat equation ut 1 u u p 1 u either on r n or on a finite ball under the dirichlet boundary conditions we assume that the exponent p is supercritical in the sobolev sense that is p ps n 2 n 2 we prove that if ps p the above range of exponent p is optimal we will also derive various fundamental estimates for blowup that hold for any p ps and regardless of type of blowup among other things we classify local profiles of type i and type ii blowups in the rescaled coordinates w e then establish useful estimates for the so called incomplete blowup which r eveal that incomplete blowup solutions belong to nice function spaces even after the blowup time c 2004 wiley periodicals inc

supersolutions for a class of nonlinear parabolic systems abstract in this paper by using scalar nonlinear parabolic equations we construct supersolutions for a class of nonlinear parabolic systems including t u u v p x t 0 t v v u q x t 0 u v 0 x t 0 u x 0 v x 0 u 0 x v 0 x x where p 0 q 0 is a possibly unbounded smooth domain in r n and both u 0 and v 0 are nonnegative and locally integrable functions in the supersolutions enable us to obtain optimal sufficient conditions for the existence of the solutions and optimal lower estimates of blow up rate of the solutions

some new constructions of quantum mds codes it is an important task to construct quantum maximum distance separable mds codes with good parameters in the present paper we provide six new classes of q ary quantum mds codes by using generalized reed solomon grs codes and hermitian construction the minimum distances of our quantum mds codes can be larger than frac q 2 1 three of these six classes of quantum mds codes have longer lengths than the ones constructed in 1 and 2 hence some of their results can be easily derived from ours via the propagation rule moreover some known quantum mds codes of specific lengths can be seen as special cases of ours and the minimum distances of some known quantum mds codes are also improved as well

single point blow up for a semilinear initial value problem abstract the initial value problem on r r is considered u t t x u xx t x u t x u t r 0 u 0 x x where 0 and is a fixed large number it is known that for some initial values the solution u t x exists only up to some finite time t and that u t as t t for the specific initial value k where 0 xx 0 r 0 k is sufficiently large it is shown that if x 0 then lim t t u t x and lim t t u x t x exist and are finite in other words blow up occurs only at the point x 0