existence of sign changing solutions for a nonlinear elliptic problem on rn with a hardy term abstract let n 3 1 p n 2 n 2 and 0 n 2 2 4 we study the existence of multiple sign changing solutions of u h 1 r n u u u x 2 u p 1 u in r n which are not radially symmetric

elastic net oriented to fuzzy semiparametric regression model with fuzzy explanatory variables and fuzzy responses in the multivariate linear regression model it is desirable to include the important explanatory variables to achieve maximal prediction in this context the present paper is an attempt to extend the conventional elastic net multiple linear regression model adopted with a semiparametric method to fuzzy predictors and responses for this purpose kernel smoothing and elastic net penalized methods were combined to construct a novel variable selection method in a fuzzy multiple regression model some common goodness of fit criteria were also included to examine the performance of the proposed method the effectiveness of the proposed method was illustrated through three numerical examples including a simulation study and two practical cases the proposed method was also compared with several common fuzzy multiple regression models the numerical results clearly indicated that the proposed method is capable of providing sufficiently accurate results in cases where noninformative explanatory variables are removed from the model

the hybrid fuzzy least squares regression approach to modeling manufacturing processes uncertainty in manufacturing processes is caused both by randomness as in material properties and by fuzziness as in the inexact knowledge previous research has seldom considered these two types of uncertainty when modeling manufacturing processes in this paper a hybrid fuzzy least squares regression hflsr approach to modeling manufacturing processes which does take into consideration these two types of uncertainty is proposed and described and a new form of weighted fuzzy arithmetic is introduced to develop the hybrid fuzzy least squares regression method the proposed hflsr approach not only features the capability of dealing with the two types of uncertainty but also addresses the consideration of replication of responses in experiments to investigate the effectiveness of the proposed approach to process modeling it was applied to the modeling solder paste dispensing process modeling results were compared with those based on statistical regression and fuzzy linear regression it was found that the accuracy of prediction based on the hflsr is slightly better than that based on statistical regression and much better than that based on the peters fuzzy regression

fuzzy nonlinear regression with fuzzified radial basis function network a fuzzified radial basis function network frbfn is a kind of fuzzy neural network that is obtained by direct fuzzification of the well known neural model rbfn a frbfn contains fuzzy weights and can handle fuzzy in fuzzy out data this paper shows that a frbfn can also be interpreted as a kind of fuzzy expert system hence it owns the advantages of simple structure and clear physical meaning some metrics for fuzzy numbers have been extended to the metrics for n dimensional fuzzy vectors which are applicable to computations in frbfns the corresponding metric spaces for n dimensional fuzzy vectors are proved to be complete further frbfns are proved to be able to act as universal function approximators for any continuous fuzzy function defined on a compact set this paper applies the proposed frbfn to nonparametric fuzzy nonlinear regression problems for multidimensional lr type fuzzy data fuzzy nonlinear regression with frbfns can be formulated as a nonlinear mathematical programming problem two training algorithms are proposed to quickly solve the two types of problems under different criteria and constraint conditions namely the two stage and bp back propagation training algorithms simulation studies are carried out to verify the feasibility and demonstrate the advantages of the proposed approaches

the brezis nirenberg problem for the laplacian with a singular drift in rn and sn abstract we consider the brezis nirenberg problem for the laplacian with a singular drift for a geodesic ball in both r n and s n 3 n 5 the singular drift we consider derives from a potential which is symmetric around the center of the geodesic ball here the potential is given by a parameter say times the logarithm of the distance to the center of the ball in both cases we determine the exact region in the parameter space for which positive smooth solutions of this problem exist and the exact region for which there are no solutions the parameter space is characterized by the geodesic radius of the ball and the coupling constant of the linear term of the brezis nirenberg problem

a linear regression model using triangular fuzzy number coefficients fuzzy regression analysis using fuzzy linear models with symmetric triangular fuzzy number coefficients has been formulated earlier the goal of this regression is to find the coefficients of a proposed model for all given input output data sets in this paper we extend the results of a fuzzy linear regression model that uses symmetric triangular coefficient to one with non symmetric fuzzy triangular coefficients this work eradicates the inflexibility of existing fuzzy linear regression models

positive solutions for elliptic equations involving critical sobolev exponents and hardy terms with neumann boundary conditions abstract in this paper neumann problem for nonlinear elliptic equations with critical sobolev exponents and hardy terms is studied by variational method based on the variant of the mountain pass theorem of ambrosetti and rabinowitz without ps condition we prove the existence of positive solutions

a robust varying coefficient approach to fuzzy multiple regression model abstract the varying coefficient models are powerful tools for exploring the dynamic pattern between a response and a group of predictors in multiple regression models in addition robust regression is another solid approach in the regression analyses for cases whose data are contaminated with outliers or influential observations this paper proposed a novel varying coefficient model with exact predictors and fuzzy responses which can be used in cases where outliers occur in the data set for this purpose a locally weighted approximation idea and a popular m estimator were combined to estimate unknown fuzzy nonparametric varying coefficients some common goodness of fit criteria including an outlier detection criterion were also applied to examine the performance of the proposed method the effectiveness of the presented method was then illustrated through two numerical examples including a simulation study it was also compared with several common fuzzy multiple regression models the numerical results clearly indicate that the proposed method is not sensitive to the outliers moreover compared to the available fuzzy multiple regressions with constant coefficients the proposed fuzzy varying coefficient model managed to provide more accurate results

solutions of the quasilinear elliptic problem with a critical sobolev hardy exponent and a hardy type term abstract in the present paper the quasilinear elliptic problem with a critical sobolev hardy exponent and a hardy type term is considered by means of a variational method the existence of nontrivial solutions to the problem is obtained

estimating the functional relationships for quality function deployment under uncertainties product planning is one of four important processes in new product development npd using quality function deployment qfd which is a widely used customer driven approach in our opinion the first problem to be solved is how to incorporate both qualitative and quantitative information regarding relationships between customer requirements crs and engineering characteristics ecs as well as those among ecs into the problem formulation owing to the typical vagueness or imprecision of functional relationships in a product product planning is becoming more difficult particularly in a fuzzy environment in this paper an asymmetric fuzzy linear regression approach is proposed to estimate the functional relationships for product planning based on qfd firstly by integrating the least squares regression into fuzzy linear regression a pair of hybrid linear programming models with asymmetric triangular fuzzy coefficients are developed to estimate the functional relationships for product planning under uncertainties secondly using the basic concept of fuzzy regression asymmetric triangular fuzzy coefficients are extended to asymmetric trapezoidal fuzzy coefficients and another pair of hybrid linear programming models with asymmetric trapezoidal fuzzy coefficients is proposed the main advantage of these hybrid programming models is to integrate both the property of central tendency in least squares and the possibilistic property in fuzzy regression next the illustrated example shows that trapezoidal fuzzy number coefficients have more flexibility to handle a wider variety of systematic uncertainties and ambiguities that cannot be modeled efficiently using triangular number fuzzy coefficients both asymmetric triangular and trapezoidal fuzzy number coefficients can be applicable to a much wider variety of design problems where uncertain qualitative and fuzzy relationships are involved than when symmetric triangular fuzzy numbers are used finally future research direction is also discussed

nodal and positive solutions for singular semilinear elliptic equations with critical exponents in symmetric domains abstract in this paper we consider the semilinear elliptic problem u u x 2 u k x u 2 2 u in u 0 on where is a smooth bounded domain in r n n 4 0 n 2 2 4 2 2 n n 2 is the critical sobolev exponent k x is a continuous function when and k x are invariant under a group of orthogonal transformations we prove the existence of nodal and positive solutions for 0 1 where 1 is the first dirichlet eigenvalue of x 2 on

multiple positive solutions for semilinear elliptic equations involving multi singular inverse square potentials and concave convex nonlinearities abstract in this paper we deal with the existence and multiplicity of positive solutions for the multi singular semilinear elliptic equations u i 1 k i x a i 2 u u 2 2 u u q 2 u x where r n n 3 is a smooth bounded domain such that the points a i i 1 2 k k 2 are different 0 i n 2 2 2 0 1 q 2 and 2 2 n n 2 the results depend crucially on the parameters q and i for i 1 2 k

solutions to critical elliptic equations with multi singular inverse square potentials abstract let be an open bounded domain in r n n 3 with smooth boundary we are concerned with the multi singular critical elliptic problem u i 1 k i u x a i 2 q x u 2 2 u u h 0 1 where i r 2 2 n n 2 a i 1 i k and q x is a positive bounded function on using moser iteration we prove the asymptotic behavior of solutions for at points a i 1 i k by exploiting the effect of the coefficient of the critical nonlinearity we by means of a variational method establish the existence of positive and sign changing solutions for problem

fuzzy regression models using the least squares method based on the concept of distance fuzzy regression models are developed to construct the relationship between explanatory variables and responses in a fuzzy environment in order to increase the explanatory performance of the model the least squares method is applied to determine the numeric coefficients based on the concept of distance unlike most existing approaches the numeric coefficients in the proposed model can have negative values the proposed model minimizes total estimation error in terms of the sum of the average squared distance between the observed and estimated responses based on a few alpha cuts the proposed approach is not limited to triangular fuzzy numbers it can be used to carry out a large number of fuzzy observations efficiently because the model is based on traditional statistical methods comparisons with existing methods show that based on the total estimation error using the mean squared error and kim and bishu s criterion the explanatory performance of the proposed model is satisfactory

a partial robust ridge based regression model with fuzzy predictors responses abstract this paper extends the conventional semi parametric partial linear regression model with fuzzy predictors and fuzzy response in cases where 1 outliers occur in data set 2 multicollinearity exists in predictors for this purpose the classical kernel fitting method is first extended for estimating unknown fuzzy smooth function then inspired by the conventional local linear smoothing method a kernel based weighted criterion is suggested for determining the exact coefficients on the basis of ridge methodology some common goodness of fit criteria are also used to examine the performance of the proposed method the effectiveness of the proposed method is then illustrated through two numerical examples including a simulation study the proposed method is also compared with some common fuzzy multiple regression models with fuzzy predictors and fuzzy responses the numerical results clearly indicate that 1 the proposed method is not sensitive to the outliers 2 the proposed method provides sufficiently accurate results in cases where multicollinearity happens among predictors and 3 comparing to existing fuzzy multiple regressions the proposed model provides more efficient predictions

existence of solution for a singular critical elliptic equation in this paper a singular semilinear elliptic problem involving the critical sobolev exponent is studied by variational method the existence of a solution is proved under certain conditions the hardy inequality is used and plays an important role in the discussion

on the existence and multiplicity of solutions for a class of singular elliptic problems abstract in this paper we prove the existence and multiplicity of solutions for a class of elliptic problems with singular coefficients we also prove the existence of a sign changing solution for a brezis nirenberg type problem with hardy potential and singular coefficients our proofs are based on the krasnoselskii genus the concentration compactness principle of lions and the linking theorem

sign changing solutions for elliptic problems with critical sobolev hardy exponents abstract let r n be a smooth bounded domain such that 0 n 7 0 s 2 s 2 n s n 2 we prove the existence of sign changing solutions for the singular critical problem u u x 2 u 2 s 2 x s u u with dirichlet boundary condition on for suitable positive parameters and

least squares estimates in fuzzy regression analysis abstract regression is a very powerful methodology for forecasting which is considered as an essential component of successful or applications in this paper an idea stemmed from the classical least squares is proposed to handle fuzzy observations in regression analysis based on the extension principle the membership function of the sum of squared errors is constructed the fuzzy sum of squared errors is a function of the regression coefficients to be determined which can be minimized via a nonlinear program formulated under the structure of the chen klein method for ranking fuzzy numbers to illustrate how the proposed method is applied three cases one crisp input fuzzy output one fuzzy input fuzzy output and one non triangular fuzzy observations are exemplified the results show that the least squares method of this paper is able to determine the regression coefficients with better explanatory power most important it works for all types of fuzzy observations not restricted to the triangular one

interaction between the geometry of the boundary and positive solutions of a semilinear neumann problem with critical nonlinearity abstract we consider the problem u u un 2 n 2 u 0 in u v 0 on where is a bounded smooth domain in r n n 3 we show that for large least energy solutions of the above problem have a unique maximum point p on and the limit points of p as are contained in the set of the points of maximum mean curvature we also prove that if has k peaks then the equation has at least k solutions for large

existence of solutions for a nonlinear pde with an inverse square potential abstract via linking theorem and delicate energy estimates the existence of nontrivial solutions for a nonlinear pde with an inverse square potential and critical sobolev exponent is proved this result gives a partial positive answer to an open problem proposed in ferrero and gazzola j differential equations 177 2001 494

trace minimization and definiteness of symmetric pencils abstract a symmetric matrix pencil a b of order n is called positive definite if there is a such that the matrix a b is positive definite we consider the case with b nonsingular and show that the definiteness is closely related to the existence of min tr x t ax under the condition x t bx j 1 where j 1 is a given diagonal matrix of order n and j 2 1 i we also prove an analog of the cauchy interlacing theorem for some such pencils

elliptic problems with critical exponents and hardy potentials this paper is devoted to the existence of positive solutions of a dirichlet problem with critical exponent and a singular potential under various assumption on the domain omega which include some kinds of unbounded domains we prove the existence of ground states and of symmetric solutions c 2002 elsevier science usa all rights reserved

theoretical choice of the optimal threshold for possibilistic linear model with noisy input based on possibility concepts various possibilistic linear models plms have been proposed and their pivotal role in fuzzy modeling and associated applications has been established when adopting plms one has to adopt an appropriate threshold lambda value however choosing such a value is by no means trivial and is still an open theoretical issue in this paper we propose a solution by first extending the plm to its regularized version i e a regularized plm rplm such that its generalization capability can be enhanced the rplm is then formulated as a maximum a posteriori map framework which facilitates the determination of the theoretically optimal threshold value for the rplm with noisy input our mathematical derivations reveal the approximately inversely proportional relationship between the threshold and the standard deviation of gaussian noisy input this is also confirmed by the simulation results this finding is very helpful for the practical applications of both plms and rplms

on computing the generalized crawford number of a matrix abstract we use methods of geometric computing combined with hermitean matrix eigenvalue eigenvector evaluations to find the generalized crawford number of a pair of hermitean matrices s 1 and s 2 quickly and to high precision the classical crawford number is defined as the minimal distance from zero in c to the field of values of the associated complex matrix a s 1 is 2 if zero lies outside the field of values of a we describe test and compare a geometry based matlab code for finding for the generalized crawford number that measures the smallest distance of zero from the boundary of the field of values of a even if zero lies inside

weak eigenfunctions for the linearization of extremal elliptic problems abstract we consider the semilinear elliptic problem formula where is a nonnegative parameter and g is a positive nondecreasing convex nonlinearity there exists a value of the parameter which is extremal in terms of existence of solution we study the linearization of the semilinear problem at the extremal weak solution corresponding to the parameter in some cases this linearized problem has discrete and positive h 1 0 spectrum however we prove that there always exists a positive weak eigenfunction in l 1 with eigenvalue zero for this linearized problem the zero l 1 eigenvalue is coherent with the nonexistence of solutions of the semilinear problem for finally we find all weak eigenfunctions and eigenvalues for the linearization of the extremal problem when is the unit ball and g u e u or g u 1 u p

a fuzzy linear regression model with better explanatory power previous studies on fuzzy linear regression analysis have a common characteristic of increasing spreads for the estimated fuzzy responses as the independent variable increases its magnitude which is not suitable for general cases this paper proposes a two stage approach to construct the fuzzy linear regression model in the first stage the fuzzy observations are defuzzified so that the traditional least squares method can be applied to find a crisp regression line showing the general trend of the data in the second stage the error term of the fuzzy regression model which represents the fuzziness of the data in a general sense is determined to give the regression model the best explanatory power for the data the results from two examples one with crisp data and the other with fuzzy data for the independent variable indicate that the two stage method proposed in this paper has better performance than the previous studies

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

positive solutions for neumann elliptic problems involving critical hardy sobolev exponent with boundary singularities abstract in this paper we consider the existence of positive solution for some semilinear elliptic equations with neumann boundary condition involving a critical hardy sobolev exponent and hardy terms with boundary singularities using mountain pass lemma without ps condition and the strong maximum principle we get the existence of a positive solution

detecting a definite hermitian pair and a hyperbolic or elliptic quadratic eigenvalue problem and associated nearness problems an important class of generalized eigenvalue problems ax lambdabx is those in which a and b are hermitian and some real linear combination of them is definite for the quadratic eigenvalue problem qep lambda 2 a lambdab c x 0 with hermitian a b and c and positive definite a particular interest focuses on problems in which x bx 2 4 x ax x cx is one signed for all non zero x for the positive sign these problems are called hyperbolic and for the negative sign elliptic the important class of overdamped problems arising in mechanics is a sub class of the hyperbolic problems for each of these classes of generalized and quadratic eigenvalue problems we show how to check that a putative member has the required properties and we derive the distance to the nearest problem outside the class for definite pairs a b the distance is the crawford number and we derive bisection and level set algorithms both for testing its positivity and for computing it testing hyperbolicity of a qep is shown to reduce to testing a related pair for definiteness the distance to the nearest non hyperbolic or non elliptic n x n qep is shown to be the solution of a global minimization problem with n 1 dearees of freedom numerical results are given to illustrate the theory and algorithms c 2002 published by elsevier science inc

a perturbation bound for definite pencils we attempt to generalize a well known result on spectral variations of a hermitian matrix due to mirsky to the definite generalized eigenvalue problem we also point out that some results on perturbations of definite pencils due to stewart can be slightly improved

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

hermitian matrix polynomials with real eigenvalues of definite type part i classification the spectral properties of hermitian matrix polynomials with real eigenvalues have been extensively studied through classes such as the definite or definitizable pencils definite hyperbolic or quasihyperbolic matrix polynomials and overdamped or gyroscopically stabilized quadratics we give a unified treatment of these and related classes that uses the eigenvalue type or sign characteristic as a common thread equivalent conditions are given for each class in a consistent format we show that these classes form a hierarchy all of which are contained in the new class of quasidefinite matrix polynomials as well as collecting and unifying existing results we make several new contributions we propose a new characterization of hyperbolicity in terms of the distribution of the eigenvalue types on the real line by analyzing their effect on eigenvalue type we show that homogeneous rotations allow results for matrix polynomials with nonsingular or definite leading coefficient to be translated into results with no such requirement on the leading coefficient which is important for treating definite and quasidefinite polynomials we also give a sufficient condition for a quasihyperbolic matrix polynomial to be diagonalizable by structure preserving congruence and show that this condition is always satisfied in the quadratic case and for any hyperbolic matrix polynomial thereby identifying an important new class of diagonalizable matrix polynomials

a note on the sign changing solutions to elliptic problems with critical sobolev and hardy terms let rn be a bounded domain such that 0 n 7 2 2nn 2 we obtain existence of sign changing solutions for the dirichlet problem u u x 2 u 2 2u u on u 0 on for suitable positive numbers and

existence of positive solutions for a class of degenerate or singular equations abstract in this paper we consider the class of equations div x 2 a u a x 2 a u u x b p x r n 0 where a b r n 2 a r 0 and 0 p 2 n n 2 1 a b in various range of the parameters involved we obtain both results of existence and nonexistence of positive solutions by combining the mountain pass lemma and the pohozaev type identity

multiple regression with fuzzy data in this paper we propose an iterative algorithm for multiple regression with fuzzy variables while using the standard least squares criterion as a performance index we pose the regression problem as a gradient descent optimisation the separation of the evaluation of the gradient and the update of the regression variables makes it possible to avoid undue complication of analytical formulae for multiple regression with fuzzy data the origins of fuzzy input data are traced back to the fundamental concept of information granulation and an example fcm based granulation method is proposed and illustrated by some numerical examples the proposed multiple regression algorithm is applied to one three and nine dimensional synthetic data sets as well as the 13 dimensional boston housing dataset from the machine learning repository the algorithm s performance is illustrated by the corresponding plots of convergence of regression parameters and the values of the prediction error of the resulting regression model general comments on the numerical complexity of the proposed algorithm are also provided

solutions for a singular critical growth problem with a weight in this paper we consider some semilinear elliptic equations with hardy potential by using linking theorem in p rabinowitz minimax methods in critical points theory with applications to differential equations cbms reg conf ser math vol 65 amer math soc providence ri 1986 and analyzing the effect of nonlinearities we establish the existence of nontrivial solutions

existence of multiple positive solutions to singular elliptic systems involving critical exponents abstract this paper is devoted to investigate multiple positive solutions to a singular elliptic system where the nonlinearity involves a combination of concave and convex terms by exploiting the effect of the coefficient of the critical nonlinearity and a variational method we establish the main result which is based on the argument of the compactness

fuzzy nonlinear regression analysis using a random weight network modeling a fuzzy in fuzzy out system where both inputs and outputs are uncertain is of practical and theoretical importance fuzzy nonlinear regression fnr is one of the approaches used most widely to model such systems in this study we propose the use of a random weight network rwn to develop a fnr model called fnrrwn where both the inputs and outputs are triangular fuzzy numbers unlike existing fnr models based on back propagation bp and radial basis function rbf networks fnrrwn does not require iterative adjustment of the network weights and biases instead the input layer weights and hidden layer biases of fnrrwn are selected randomly the output layer weights for fnrrwn are calculated analytically based on a derived updating rule which aims to minimize the integrated squared error between cut sets that correspond to the predicted fuzzy outputs and target fuzzy outputs respectively in fnrrwn the integrated squared error is solved approximately by riemann integral theory the experimental results show that the proposed fnrrwn method can effectively approximate a fuzzy in fuzzy out system fnrrwn obtains better prediction accuracy in a lower computational time compared with existing fnr models based on bp and rbf networks

existence of solutions for elliptic equations with critical sobolev hardy exponents let omega subset of rn be a smooth bounded domain such that 0 is an element of omega n greater than or equal to 3 0 less than or equal to s 2 2 s 2 n s n 2 we prove the existence of nontrival solutions for the singular critical problem

multiple positive solutions for robin problem involving critical weighted hardy sobolev exponents with boundary singularities abstract this paper deals with the existence of positive solutions for robin elliptic problems involving critical weighted hardy sobolev exponents with boundary singularities using the caffarelli kohn nirenberg inequalities and variational methods we prove the existence and multiplicity of positive solutions

existence and multiplicity of positive solutions for elliptic equation with critical weighted hardy sobolev exponents and boundary singularities abstract we study a semilinear elliptic equation involving critical weighted hardy sobolev exponents with boundary singularities the existence and multiplicity of positive solutions are established our method relies upon ekeland s variational principle and mountain pass lemma

positive definite combination of symmetric matrices this correspondence presents a new algorithm to find a positive definite combination of some given real symmetric matrices when this problem has solutions we show that it can be written as an optimization problem an iterative algorithm using a relaxation type method is proposed to solve this optimization problem we also prove that this algorithm has a finite step global convergence property a comparison by computer simulations shows that our algorithm is much faster than another one generally proposed in literature to find such a combination

solutions for semilinear elliptic equations with critical exponents and hardy potential abstract in this paper we answer affirmatively an open problem cf theorem 4 in ferrero and gazzola j differential equations 177 2001 494 let 0 be an open bounded domain r n n 5 and assume that 0 n 2 2 2 n 2 n 2 then for all 0 there exists a nontrivial solution with critical level in the range 0 1 n s n 2 for the problem u u x 2 u u 2 2 u in u 0 on

infinitely many solutions to perturbed elliptic equations abstract a new version of perturbation theory is developed which produces infinitely many sign changing critical points for uneven functionals the abstract result is applied to the following elliptic equations with a hardy potential and a perturbation from symmetry u u x 2 f x u p x u in u 0 on and u u q 2 x s u p x u in u 0 on where 0 s 2 is a smooth bounded domain of r n and p x u is not odd in u infinitely many sign changing solutions are obtained

finding a positive definite linear combination of two hermitian matrices abstract let a and b be n by n hermitian matrices over the complex field a result of au yeung 1 and stewart 8 states that if x a ib x 0 for all nonzero n vectors x then there is a linear combination of a and b which is positive definite in this article we present an algorithm which finds such a linear combination in a finite number of steps we also discuss the implementation of the algorithm in case a and b are real symmetric sparse matrices

expansion formulas for the inertias of hermitian matrix polynomials and matrix pencils of orthogonal projectors abstract this paper gives a group of expansion formulas for the inertias of hermitian matrix polynomials a a 2 i a 2 and a a 3 through some congruence transformations for block matrices where a is a hermitian matrix then the paper derives various expansion formulas for the ranks and inertias of some matrix pencils generated from two or three orthogonal projectors and hermitian unitary matrices as applications the paper establishes necessary and sufficient conditions for many matrix equalities to hold as well as many inequalities in the lowner partial ordering to hold

four solutions of an inhomogeneous elliptic equation with critical exponent and singular term abstract in this paper we prove the existence of four nontrivial solutions of u x 2 u u 2 2 u x 2 u f x x 0 and show that at least one of them is sign changing our results extend some previous works such as g tarantello multiplicity results for an inhomogeneous neumann problem with critical exponent manuscripta math 81 1993 51 78 d kang and y deng multiple solutions for inhomogeneous elliptic problems involving critical sobolev hardy exponents j math anal appl 60 2005 729 753 n hirano and n shioji a multiplicty result including a sign changing solution for an inhomogeneous neumann problem with critical exponent proc roy soc edinburgh 137a 2007 333 347

existence and multiplicity of positive solutions for some hardy sobolev critical elliptic equation with boundary singularities abstract in this paper by ekeland s variational principle and strong maximum principle we consider the existence and multiplicity of positive solutions for some semilinear elliptic equation involving critical hardy sobolev exponents and hardy terms with boundary singularities

the nearest definite pair for the hermitian generalized eigenvalue problem the generalized eigenvalue problem ax lambda bx has special properties when a b is a hermitian and definite pair given a general hermitian pair a b it is of interest to find the nearest definite pair having a specified crawford number delta 0 we solve the problem in terms of the inner numerical radius associated with the field of values of a ib we show that once the problem has been solved it is trivial to rotate the perturbed pair a da b db to a pair widetilde a widetilde b for which lambda min widetilde b achieves its maximum value delta which is a numerically desirable property when solving the eigenvalue problem by methods that convert to a standard eigenvalue problem by inverting b numerical examples are given to illustrate the analysis

the effect of domain topology on the number of positive solutions for singular elliptic problems involving the caffarelli kohn nirenberg inequalities abstract in this paper we consider the critical singular equation involving the caffarelli kohn nirenberg inequalities of the type p 1 div x 2 a u u x 2 1 a u q 2 u x d d u p 2 u x b p x u 0 x u 0 x here is a bounded domain with smooth boundary in r n and contains 0 in its interior 0 a 2 n 2 2 2 n 3 a b a 1 a d a 1 p p a b 2 n n 2 1 a b d d a d 2 n n 2 1 a d is a positive parameter and 2 q d by lusternik schnirelmann category theory we prove that the problem p 1 has at least cat positive solutions

dependency between degree of fit and input noise in fuzzy linear regression using non symmetric fuzzy triangular coefficients fuzzy linear regression flr model can be thought of as a fuzzy variation of classical linear regression model it has been widely studied and applied in diverse fields when noise exists in data it is a very meaningful topic to reveal the dependency between the parameter h i e the threshold value used to measure degree of fit in flr model and the input noise in this paper the flr model is first extended to its regularized version i e regularized fuzzy linear regression rflr model so as to enhance its generalization capability then rflr model is explained as the corresponding equivalent maximum a posteriori map problem finally the general dependency relationship that the parameter h with noisy input should follow is derived particular attention is paid to the regression model using non symmetric fuzzy triangular coefficients it turns out that with the existence of typical gaussian noisy input the parameter h is inversely proportional to the input noise our experimental results here also confirm this theoretical claim obviously this theoretical result will be helpful to make a good choice for the parameter h and to apply flr techniques effectively in practical applications

multiple solutions for inhomogeneous elliptic problems involving critical sobolev hardy exponents abstract let r n be a smooth bounded domain such that 0 n 3 0 s 2 2 s 2 n s n 2 is the critical sobolev hardy exponent f x is a given function by using the ekeland s variational principle and the mountain pass lemma we prove the existence of multiple solutions for the singular critical inhomogeneous problem u u x 2 u 2 s 2 x s u u f x with dirichlet boundary condition on under some assumptions on f x and

multiple positive solutions for semilinear elliptic equations with critical weighted hardy sobolev exponents abstract the existence and multiplicity of positive solutions are obtained for a class of semilinear elliptic equations with critical weighted hardy sobolev exponents and the concave convex nonlinearity by variational methods and some analysis techniques

quasilinear elliptic problems with critical exponents and hardy terms abstract let 0 be an open bounded domain r n n p 2 we are concerned with the multiplicity of positive solutions of p u u p 2 u x p u p 2 u q x u p 2 u u w 0 1 p where p u div u p 2 u 1 p n p np n p 0 n p p p 0 and q x is a nonnegative function on by investigating the effect of the coefficient of the critical nonlinearity we by means of variational method prove the existence of multiple positive solutions

multiplicity result for a class of elliptic equations with singular term abstract we consider the existence of nontrivial solutions of the equation u x 2 u u 2 2 u x 2 u f x u x 0 u h 0 1 where 0 is a smooth bounded domain in r n n 3 by variational methods and nehari set techniques we show that this problem under some additional hypotheses on 0 0 0 0 1 and f l has four nontrivial solutions in h 0 1 and that least one of them is sign changing

existence and multiplicity of positive solutions for a class of semilinear elliptic equations involving hardy term and hardy sobolev critical exponents abstract existence and multiplicity of positive solutions are studied for a class of semilinear elliptic equations with hardy term hardy sobolev critical exponents and sublinear nonlinearity by variational methods and some analysis techniques

fuzzy rough regression with application to wind speed prediction accurate wind speed prediction is a prerequisite of large scale wind power generation there are several uncertain factors which degrade the performance of the current wind speed prediction systems fuzzy rough sets are considered as a powerful tool to deal with uncertainty and have been widely discussed and applied in classification learning in this work we describe a regression algorithm based on fuzzy rough sets consisting of fuzzy partition fuzzy approximation and estimation of regression values in this algorithm the training set is divided into k fuzzy classes with fuzzy partition and then the predicted values of test samples are determined in the finite intervals with fuzzy rough approximation finally they are estimated with lower and upper limits of the intervals numerical experiments on uci data sets and wind speed prediction show the effectiveness of the proposed algorithm

eigenvalue based algorithm and analysis for nonconvex qcqp with one constraint a nonconvex quadratically constrained quadratic programming qcqp with one constraint is usually solved via a dual sdp problem or more s algorithm based on iteratively solving linear systems in this work we introduce an algorithm for qcqp that requires finding just one eigenpair of a generalized eigenvalue problem and involves no outer iterations other than the usually black box iterations for computing the eigenpair numerical experiments illustrate the efficiency and accuracy of our algorithm we also analyze the qcqp solution extensively including difficult cases and show that the canonical form of a matrix pair gives a complete classification of the qcqp in terms of boundedness and attainability and explain how to obtain a global solution whenever it exists