an evidential reasoning based consensus model for multiple attribute group decision analysis problems with interval valued group consensus requirements abstract with the aim of modeling multiple attribute group decision analysis problems with group consensus gc requirements a gc based evidential reasoning approach and further an attribute weight based feedback model are sequentially developed based on an evidential reasoning er approach in real situations however giving precise crisp assessments for alternatives is often too restrictive and difficult for experts due to incompleteness or lack of information experts may also find it difficult to give appropriate assessments on specific attributes due to limitation or lack of knowledge experience and provided data about the problem domain in this paper an er based consensus model ercm is proposed to deal with these situations in which experts assessments are interval valued rather than precise correspondingly predefined interval valued gc igc requirements need to be reached after group analysis and discussion within specified times also the process of reaching igc is accelerated by a feedback mechanism including identification rules at three levels consisting of the attribute alternative and global levels and a suggestion rule particularly recommendations on assessments in the suggestion rule are constructed based on recommendations on their lower and upper bounds detected by the identification rule at a specific level a preferentially developed industry selection problem is solved by the ercm to demonstrate its detailed implementation process validity and applicability

global solutions to the two dimensional quasi geostrophic equation with critical or super critical dissipation the two dimensional quasi geostrophic 2d qg equation with critical and super critical dissipation is studied in sobolev space h s 2 for critical case open image in new window existence of global large solutions in h s is proved for s open image in new window when open image in new window is small this generalizes and improves the results of constantin d cordoba and wu 4 for s 1 2 and the result of a cordoba and d cordoba 8 for s open image in new window for s 1 these solutions are also unique the improvement for pushing s down from 1 to open image in new window is somewhat surprising and unexpected for super critical case 0 open image in new window existence and uniqueness of global large solution in h s is proved when the product open image in new window is small for suitable s 2 2 p 1 and 0 1

generalized surface quasi geostrophic equations with singular velocities this paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals the first family is a generalized surface quasigeostrophic sqg equation with the velocity field u related to the scalar by where and is the zygmund operator the borderline case 1 corresponds to the sqg equation and the situation is more singular for 1 we obtain the local existence and uniqueness of classical solutions the global existence of weak solutions and the local existence of patch type solutions the second family is a dissipative active scalar equation with which is at least logarithmically more singular than the velocity in the first family we prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data this result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by k ohkitani 2012 wiley periodicals inc

the evidential reasoning approach for multiple attribute decision analysis using interval belief degrees abstract multiple attribute decision analysis mada problems having both quantitative and qualitative attributes under uncertainty can be modeled using the evidential reasoning er approach several types of uncertainties such as ignorance and fuzziness can be modeled in the er framework in this paper the er approach will be extended to model new types of uncertainties including interval belief degrees and interval data that could be incurred in decision situations such as group decision making the dempster shafer d s theory of evidence is first extended which is one of the bases of the er approach the analytical er algorithm is used to combine all evidence simultaneously two pairs of nonlinear optimization models are constructed to estimate the upper and lower bounds of the combined belief degrees and to compute the maximum and the minimum expected utilities of each alternative respectively interval data are equivalently transformed to interval belief degrees and are incorporated into the nonlinear optimization models a cargo ship selection problem is examined to show the implementation process of the proposed approach

global well posedness for a modified critical dissipative quasi geostrophic equation in this paper we consider the following modified quasi geostrophic equation t u d 0 u d 1r x r2 with 0 and 0 1 1 2 when 0 1 the equation was firstly introduced by constantin iyer and wu 2008 in 11 here by using the modulus of continuity method we prove the global well posedness of the system as a byproduct we also show that for every 0 2 the lipschitz norm of the solution has a uniform exponential upper bound

a new bernstein s inequality and the 2d dissipative quasi geostrophic equation we show a new bernstein s inequality which generalizes the results of cannone planchon danchin and lemarie rieusset as an application of this inequality we prove the global well posedness of the 2d quasi geostrophic equation with the critical and super critical dissipation for the small initial data in the critical besov space and local well posedness for the large initial data

the owa based consensus operator under linguistic representation models using position indexes when using linguistic approaches to solve decision problems we need linguistic representation models the symbolic model the 2 tuple fuzzy linguistic representation model and the continuous linguistic model are three existing linguistic representation models based on position indexes together with these three linguistic models the corresponding ordered weighted averaging operators such as the linguistic ordered weighted averaging operator the 2 tuple ordered weighted averaging operator and the extended ordered weighted averaging operator have been developed respectively in this paper we analyze the internal relationship among these operators and propose a consensus operator under the continuous linguistic model or the 2 tuple fuzzy linguistic representation model the proposed consensus operator is based on the use of the ordered weighted averaging operator and the deviation measures some desired properties of the consensus operator are also presented in particular the consensus operator provides an alternative consensus model for group decision making this consensus model preserves the original preference information given by the decision makers as much as possible and supports consensus process automatically without moderator

infinite energy solutions of the surface quasi geostrophic equation abstract we study the formation of singularities of a 1d non linear and non local equation we show that this equation provides solutions of the surface quasi geostrophic equation with infinite energy the existence of self similar solutions and the blow up for classical solutions are shown

an outranking based general approach to solving group multi objective optimization problems abstract this paper presents a general approach to solving multi objective programming problems with multiple decision makers the proposal is based on optimizing a bi objective measure of collective satisfaction group satisfaction is understood as a reasonable balance between the strengths of an agreeing and an opposing coalition considering also the number of decision makers not belonging to any of these coalitions accepting the vagueness of collective satisfaction even the vagueness of person satisfaction fuzzy outranking relations and other fuzzy logic models are used our method transforms a group multi objective optimization problem into a group choice problem on a decision set composed of a relatively small set of alternatives this set contains the possible acceptable consensuses in the parameter space once such a set has been identified other well known techniques can be used to reach the final choice main advantages a each individual decision maker is concerned with his her own multi objective optimization problem only sharing decision variables own constraints and own mapping between decision variables and objective space are allowed b the search for the best agreement is not limited to portions of the pareto frontiers c no voting rule is used by the optimization algorithm no to some extent arbitrary way of handling collective preferences is needed d no assumptions of transitivity and comparability of preference relations are needed and e the concepts of satisfaction non satisfaction do not depend on distance measures or other to some extent arbitrary norms very good performance of the whole proposal is illustrated by a real size example

the sharp corner formation in 2d euler dynamics of patches infinite double exponential rate of merging for the 2d euler dynamics of patches we investigate the convergence to the singular stationary solution in the presence of a regular strain it is proved that the rate of merging can be double exponential infinitely in time and the estimates we obtain are sharp

a geometrical constraint for capillary jet breakup abstract the formation of thinning filaments is commonly observed previously to the break up of a very viscous jet this paper shows that a fluid under capillary forces cannot break up through the uniform collapse of a filament

long time dynamics of forced critical sqg we prove the existence of a compact global attractor for the dynamics of the forced critical surface quasi geostrophic equation sqg and prove that it has finite fractal box counting dimension in order to do so we give a new proof of global regularity for critical sqg the main ingredient is the nonlinear maximum principle in the form of a nonlinear lower bound on the fractional laplacian which is used to bootstrap the regularity directly from l to c without the use of de giorgi techniques we prove that for large time the norm of the solution measured in a sufficiently strong topology becomes bounded with bounds that depend solely on norms of the force which is assumed to belong merely to l h1 using the fact that the solution is bounded independently of the initial data after a transient time in spaces conferring enough regularity we prove the existence of a compact absorbing set for the dynamics in h1 obtain the compactness of the linearization and the continuous differentiability of the solution map we then prove exponential decay of high yet finite dimensional volume elements in h1 along solution trajectories and use this property to bound the dimension of the global attractor

aggregation of utility based individual preferences for group decision making abstract multi attribute utility theory maut elicits an individual decision maker s preferences for single attributes and develops a utility function by mathematics formulation to add up the preferences of the entire set of attributes when assessing alternatives a common aggregation method of maut for group decisions is the simple additive weighting saw method which does not consider the different preferential levels and preferential ranks for individual decision makers assessments of alternatives in a decision group and thus seems too intuitive in achieving the consensus and commitment for group decision aggregation in this paper the preferential differences denoting the preference degrees among different alternatives and preferential priorities denoting the favorite ranking of the alternatives for each decision maker are both considered and aggregated to construct the utility discriminative values for assessing alternatives in a decision group a comparative analysis is performed to compare the proposed approach to the saw model and a satisfaction index is used to investigate the satisfaction levels of the final two resulting group decisions in addition a feedback interview is conducted to understand the subjective perceptions of decision makers while examining the results obtained from these two approaches for the second practical case both investigation results show that the proposed approach is able to achieve a more satisfying and agreeable group decision than that of the saw method

a rational consensus model in group decision making using linguistic assessments abstract a new consensus model for the consensus reaching process in a linguistic framework is presented in heterogeneous group decision making problems called rational consensus model it is guided by some linguistic consensus and linguistic consistency meaures all the measures are calculated from a set of linguistic preference relations used to provide experts opinions this consensus model allows more rational consensus solutions to be obtained and thus more human consistency to be incorporated in decision support systems

a maximum principle applied to quasi geostrophic equations we study the initial value problem for dissipative 2d quasi geostrophic equations proving local existence global results for small initial data in the super critical case decay of lp norms and asymptotic behavior of viscosity solution in the critical case our proofs are based on a maximum principle valid for more general flows

interval efficiency assessment using data envelopment analysis this paper studies how to conduct efficiency assessment using data envelopment analysis dea in interval and or fuzzy input output environments a new pair of interval dea models is constructed on the basis of interval arithmetic which differs from the existing dea models handling interval data in that the former is a linear ccr model without the need of extra variable alternations and uses a fixed and unified production frontier i e the same constraint set to measure the efficiencies of decision making units dmus with interval input and output data while the latter is usually a nonlinear optimization problem with the need of extra variable alternations or scale transformations and utilizes variable production frontiers i e different constraint sets to measure interval efficiencies ordinal preference information and fuzzy data are converted into interval data through the estimation of permissible intervals and a level sets respectively and are incorporated into the interval dea models the proposed interval dea models are developed for measuring the lower and upper bounds of the best relative efficiency of each dmu with interval input and output data which are different from the interval formed by the worst and the best relative efficiencies of each dmu a minimax regret based approach mra is introduced to compare and rank the efficiency intervals of dmus two numerical examples are provided to show the applications of the proposed interval dea models and the preference ranking approach

the group consensus based evidential reasoning approach for multiple attributive group decision analysis abstract many multiple attribute decision analysis problems include both quantitative and qualitative attributes with various kinds of uncertainties such as ignorance fuzziness interval data and interval belief degrees an evidential reasoning er approach developed in the 1990s and in recent years can be used to model these problems in this paper the er approach is extended to group consensus gc situations for multiple attributive group decision analysis problems in order to construct and check the gc a compatibility measure between two belief structures is developed first considering two experts utilities the compatibility between their assessments is naturally constructed using the compatibility measure based on the compatibility between two experts assessments the gc at a specific level that may be the attribute level the alternative level or the global level can be constructed and reached after the group analysis and discussion within specified times under the condition of gc we conduct a study on the forming of group assessments for alternatives the achievement of the aggregated utilities of assessment grades and the properties and procedure of the extended er approach an engineering project management software selection problem is solved by the extended er approach to demonstrate its detailed implementation process and its validity and applicability

nonexistence of simple hyperbolic blow up for the quasi geostrophic equation the problem we are concerned with is whether singularities form in finite time in incompressible fluid flows it is well known that the answer is no in the case of euler and navier stokes equations in dimension two in dimension three it is still an open problem for these equations in this paper we focus on a two dimensional active scalar model for the 3d euler vorticity equation constantin majda and tabak 7 suggested by studying rigorous theorems and detailed numerical experiments a general principle if the level set topology in the temperature field for the 2d quasi geostrophic active scalar in the region of strong scalar gradients does not contain a hyperbolic saddle then no finite time singularity is possible numerical simulations showed evidence of singular behavior when the geometry of the level sets of the active scalar contain a hyperbolic saddle there is a naturally associated notion of simple hyperbolic saddle breakdown the main theorem we present in this paper shows that such breakdown cannot occur in finite time we also show that the angle of the saddle cannot close in finite time and it cannot be faster than a double exponential in time using the same techniques we see that analogous results hold for incompressible 2d and 3d euler these results were announced in 9 but with a slight difference in the definition of a simple hyperbolic saddle the definition given in section 4 generalizes the one given in the announcement see also constantin 4 discussed in section 7 remark 5 below the whole work described in this paper is basically part of the author s thesis i am particularly grateful to my thesis advisor charles fefferman for his attention support guidance and advice i am indebted to d christodoulou and p constantin for helpful corrections and suggestions i wish to thank a majda for suggesting the subject and e tabak for discussions and com

a cardinal dissensus measure based on the mahalanobis distance abstract in this paper we address the problem of measuring the degree of consensus dissensus in a context where experts or agents express their opinions on alternatives or issues by means of cardinal evaluations to this end we propose a new class of distance based consensus model the family of the mahalanobis dissensus measures for profiles of cardinal values we set forth some meaningful properties of the mahalanobis dissensus measures finally an application over a real empirical example is presented and discussed

remarks on the global regularity for the super critical 2d dissipative quasi geostrophic equation abstract in this article we apply the method used in the recent elegant proof by kiselev nazarov and volberg of the well posedness of critically dissipative 2d quasi geostrophic equation to the super critical case we prove that if the initial value satisfies 0 l 1 2 s 0 l 2 s c s for some small number c s 0 where s is the power of the fractional laplacian then no finite time singularity will occur for the super critically dissipative 2d quasi geostrophic equation

geometric constrains for global regularity of 2d quasi geostrophic flows we study the two dimensional quasi geostrophic equations 2d qg in sobolev spaces we first prove a new analytic condition for global regularity which is both sufficient and necessary we then prove several new results on the geometric constraints on the 2d qg active scalar which suppress the development of singularity from the nonlinear stretching mechanism we focus mainly on the case with critical dissipation our results are also relevant to the inviscid case

finite time singularity for the modified sqg patch equation it is well known that the incompressible euler equations in two dimensions have globally regular solutions the inviscid surface quasi geostrophic sqg equation has a biot savart law which is one derivative less regular than in the euler case and the question of global regularity for its solutions is still open we study here the patch dynamics in the half plane for a family of active scalars which interpolates between these two equations via a parameter 0 12 appearing in the kernels of their biot savart laws the values 0 and 12 correspond to the 2d euler and sqg cases respectively we prove global in time regularity for the 2d euler patch model even if the patches initially touch the boundary of the half plane on the other hand for any sufficiently small 0 we exhibit initial data which lead to a singularity in finite time thus these results show a phase transition in the behavior of solutions to these equations and provide a rigorous foundation for classifying the 2d euler equations as critical department of mathematics rice university 6100 s main st houston tx 77005 1892 kiselev rice edu department of mathematics stanford university stanford ca 94305 ryzhik math stanford edu school of mathematics georgia institute of technology 686 cherry street atlanta ga 30332 0160 yaoyao math gatech edu department of mathematics university of wisconsin 480 lincoln dr madison wi 53706 1325 zlatos math wisc edu

on the conserved quantities for the weak solutions of the euler equations and the quasi geostrophic equations in this paper we obtain sufficient conditions on the regularity of the weak solutions to guarantee conservation of the energy and the helicity for the incompressible euler equations the regularity of the weak solutions are measured in terms of the triebel lizorkin type of norms dot mathcal f s p q and the besov norms dot mathcal b s p q in particular in the besov space case our results refine the previous ones due to constantin e titi energy and the author of this paper helicity where the regularity is measured by a special class of the besov space norm dot mathcal b s p infty dot mathcal n s p which is the nikolskii space we also obtain a sufficient regularity condition for the conservation of the lp norm of the temperature function in the weak solutions of the quasi geostrophic equation

nonexistence of self similar singularities for the 3d incompressible euler equations we prove that there exists no self similar finite time blowing up solution to the 3d incompressible euler equations if the vorticity decays sufficiently fast near infinity in mathbb r 3 by a similar method we also show nonexistence of self similar blowing up solutions to the divergence free transport equation in mathbb r n this result has direct applications to the density dependent euler equations the boussinesq system and the quasi geostrophic equations for which we also show nonexistence of self similar blowing up solutions

numerical solutions of the thermistor problem by spline finite elements this paper presents approximate steady state solutions of a one dimensional positive temperature coefficient ptc thermistor problem having a step function electrical conductivity that is a highly non linear function of the temperature using subdomain collocation and petrov galerkin methods based on spline finite elements the resulting system of ordinary differential equations is solved by the usual crank nicolson finite difference method using a variant of thomas algorithm it is shown that the numerical solutions obtained by the present methods exhibit the correct physical characteristics of the problem and they are in very good agreement with the exact solution a fourier stability analysis of each method is also investigated

numerical solutions of the thermistor equations abstract this paper examines heat conduction in a thermistor used as a current surge regulator the problem consists of coupled nonlinear nonlocal parabolic initial boundary value problems simplifying assumptions are made which lead to two different problems each of which consists of a one space dimensional nonlocal parabolic initial boundary value problem numerical methods for the approximate solutions of both the steady state and the transient problems are described and the results of the numerical experiments are presented

evidential reasoning based preference programming for multiple attribute decision analysis under uncertainty multiple attribute decision analysis mada problems having both quantitative and qualitative attributes under uncertainty can be modelled and analysed using the evidential reasoning er approach several types of uncertainty such as ignorance and fuzziness can be consistently modelled in the er framework in this paper both interval weight assignments and interval belief degrees are considered which could be incurred in many decision situations such as group decision making based on the existing er algorithm several pairs of preference programming models are constructed to support global sensitivity analysis based on the interval values and to generate the upper and lower bounds of the combined belief degrees for distributed assessment and also the expected values for ranking of alternatives a post optimisation procedure is developed to identify non dominated solutions examine the robustness of the partial ranking orders generated and provide guidance for the elicitation of additional information for generating more desirable assessment results a car evaluation problem is examined to show the implementation process of the proposed approach

a two dimensional model for quasigeostrophic flow comparison with the two dimensional euler flow abstract a simple two dimensional model for quasigeostrophic flow is contrasted with the two dimensional incompressible euler equations the model arises under the assumptions of fast rotation uniform stratification and uniform potential vorticity it is found that the more loca feed back of the quasigeostrophic model gives rise to strongly nonlinear front formation as opposed to two dimensional euler where the steepening process of mature fronts obeys a nonlocal nearly linear mechanism

weighted cautious conjunctive rule for belief functions combination denoeux proposed an operator called the cautious conjunctive rule ccr to combine non dogmatic belief functions i e the frame is considered a focal element from reliable dependent sources which can occur in practice however in cases such as uncertain multiple attribute and group decision analyses belief functions may denote data that vary in importance this paper extends ccr as a weighted ccr wccr to combine belief functions in consideration of their relative weights in such situations properties of wccr are analyzed proven and demonstrated using numerical examples in particular the consideration of relative weights enables wccr to combine dogmatic belief functions the normalized wccr is further constructed and used to assess the trustworthiness of a hospital information system employed in many hospitals in anhui province china

existence and uniqueness of the solution to the dissipative 2d quasi geostrophic equations in the sobolev space we study the two dimensional dissipative quasi geostrophic equations in the sobolev space open image in new window existence and uniqueness of the solution local in time is proved in h s when s 2 1 existence and uniqueness of the solution global in time is also proved in h s when s 2 1 and the initial data open image in new window is small for the case s 2 1 we also obtain the unique large global solution in h s provided that open image in new window is small enough

analytical behavior of two dimensional incompressible flow in porous media in this paper we study the analytic structure of a two dimensional mass balance equation of an incompressible fluid in a porous medium given by darcy s law we obtain local existence and uniqueness by the particle trajectory method and we present different global existence criterions these analytical results with numerical simulations are used to indicate nonformation of singularities moreover blow up results are shown in a class of solutions with infinite energy

a frequency localized maximum principle applied to the 2d quasi geostrophic equation in this paper we prove a maximum principle for a frequency localized transport diffusion equation as an application we prove the local well posedness of the supercritical quasi geostrophic equation in the critical besov spaces mathring b 1 alpha infty q and global well posedness of the critical quasi geostrophic equation in mathring b 0 infty q for all 1 q here mathring b s infty q is the closure of the schwartz functions in the norm of b s infty q

finite time singularities in a 1d model of the quasi geostrophic equation in this paper we study 1d equations with nonlocal flux these models have resemblance of the 2d quasi geostrophic equation we show the existence of singularities in finite time and construct explicit solutions to the equations where the singularities formed are shocks for the critical viscosity case we show formation of singularities and global existence of solutions for small initial data

distance based consensus models for fuzzy and multiplicative preference relations this paper proposes a distance based consensus model for fuzzy preference relations where the weights of fuzzy preference relations are automatically determined two indices an individual to group consensus index ici and a group consensus index gci are introduced an iterative consensus reaching algorithm is presented and the process terminates until both the ici and gci are controlled within predefined thresholds the model and algorithm are then extended to handle multiplicative preference relations finally two examples are illustrated and comparative analyses demonstrate the effectiveness of the proposed methods

the maximum principle and the global attractor for the dissipative 2d quasi geostrophic equations the long time behavior of the solutions to the two dimensional dissipative quasi geostrophic equations is studied we obtain a new positivity lemma which improves a previous version of a cordoba and d cordoba 10 and 11 as an application of the new positivity lemma we obtain the new maximum principle i e the decay of the solution in l p for any p 2 when f is zero as a second application of the new positivity lemma for the sub critical dissipative case with open image in new window the existence of the global attractor for the solutions in the space h s for any s 2 1 is proved for the case when the time independent f is non zero therefore the global attractor is infinitely smooth if f is this significantly improves the previous result of berselli 2 which proves the existence of an attractor in some weak sense for the case 1 the global attractor exists in h s for any s 0 and the estimate of the hausdorff and fractal dimensions of the global attractor is also available

a variety of finite difference methods to the thermistor with a new modified electrical conductivity we consider the numerical solution of a one dimensional thermistor thermo electric problem with a new modified step function electrical conductivity which is an inherently non linear function of the temperature a variety of finite difference methods are applied to solve the problem using a new modification of the step function electrical conductivity to be satisfied the physical phenomena of the problem

scalars convected by a two dimensional incompressible flow we provide a test for numerical simulations for several two dimensional incompressible flows that appear to develop sharp fronts we show that in order to have a front the velocity has to have uncontrolled velocity growth 2001 john wiley sons inc

the evidential reasoning approach for mada under both probabilistic and fuzzy uncertainties abstract many multiple attribute decision analysis mada problems are characterised by both quantitative and qualitative attributes with various types of uncertainties incompleteness or ignorance and vagueness or fuzziness are among the most common uncertainties in decision analysis the evidential reasoning er approach has been developed in the 1990s and in the recent years to support the solution of mada problems with ignorance a kind of probabilistic uncertainty in this paper the er approach is further developed to deal with mada problems with both probabilistic and fuzzy uncertainties in this newly developed er approach precise data ignorance and fuzziness are all modelled under the unified framework of a distributed fuzzy belief structure leading to a fuzzy belief decision matrix a utility based grade match method is proposed to transform both numerical data and qualitative fuzzy assessment information of various formats into the fuzzy belief structure a new fuzzy er algorithm is developed to aggregate multiple attributes using the information contained in the fuzzy belief matrix resulting in an aggregated fuzzy distributed assessment for each alternative different from the existing er algorithm that is of a recursive nature the new fuzzy er algorithm provides an analytical means for combining all attributes without iteration thus providing scope and flexibility for sensitivity analysis and optimisation a numerical example is provided to illustrate the detailed implementation process of the new er approach and its validity and wide applicability

stationary solutions to the thermistor problem abstract a mathematical model for the description of heat conduction in a conductor in the presence of joule heating is considered the problem is to solve a coupled system of partial differential equations consisting of the heat equation with joule heating as a source and the current conservation equation with temperature dependent electrical conductivity in the steady state this is a coupled system of elliptic equations subject to general boundary conditions the existence of a weak solution is proved using schauder s fixed point theorem together with elliptic estimates uniqueness of the solution is proved for sufficiently small data a nonlocal nonlinear boundary condition is considered as well and existence of a weak solution to this problem is shown together with a sufficient condition for the existence of more than one solution

an attribute weight based feedback model for multiple attributive group decision analysis problems with group consensus requirements in evidential reasoning context abstract in an evidential reasoning context a group consensus gc based approach can model multiple attributive group decision analysis problems with gc requirements the predefined gc is reached through several rounds of group analysis and discussion gad in the approach however the gad with no guidance may not be the most appropriate way to reach the predefined gc because several rounds of gad will spend a lot of time of all experts and yet cannot help them to effectively emphasize on the assessments which primarily damage the gc in this paper an attribute weight based feedback model is constructed to effectively identify the assessments primarily damaging the gc and accelerate the gc convergence considering important attributes with the weights more than or at least equal to the mean of the weights of all attributes the feedback model constructs identification rules to identify the assessments damaging the gc for the experts to renew in addition a suggestion rule is introduced to generate appropriate recommendations for the experts to renew their identified assessments the identification rules are constructed at three levels including the attribute alternative and global levels the feedback model is used to solve an engineering project management software selection problem to demonstrate its detailed implementation process its validity and applicability and its advantages compared with the gc based approach

growth of solutions for qg and 2d euler equations the work of constantin majda tabak 1 developed an analogy between the quasi geostrophic and 3d euler equations constantin majda and tabak proposed a candidate for a singularity for the quasi geostrophic equation their numerics showed evidence of a blow up for a particular initial data where the level sets of the temperature contain a hyperbolic saddle the arms of the saddle tend to close in finite time producing a sharp front numerics studies done later by ohkitani yamada 8 and constantin nie schorghofer 2 with the same initial data suggested that instead of a singularity the derivatives of the temperature were increasing as double exponential in time the study of collapse on a curve was first studied in 1 for the quasi geostrophic equation where they considered a simplified ansatz for classical frontogenesis with trivial topology at the time of collapse the scalar is discontinuous across the curve x2 f x1 with different limiting values for the temperature on each side of the front they show that under this topology the directional field remains smooth up to the collapse which contradicts the following theorem proven in 1

solutions of the 2d quasi geostrophic equation in holder spaces abstract the 2d quasi geostrophic equation t u 0 u r is a two dimensional model of the 3d hydrodynamics equations when 1 2 the issue of existence and uniqueness concerning this equation becomes difficult it is shown here that this equation with either 0 or 0 and 0 1 2 has a unique local in time solution corresponding to any initial datum in the space c r l q for r 1 and q 1