characterizations of full stability in constrained optimization this paper is mainly devoted to the study of the so called full lipschitzian stability of local solutions to finite dimensional parameterized problems of constrained optimization which has been well recognized as a very important property from the viewpoints of both optimization theory and its applications based on second order generalized differential tools of variational anal ysis we obtain necessary and sufficient conditions for fully stable local minimizers in general classes of constrained optimization problems including problems of composite optimization mathemati cal programs with polyhedral constraints as well as problems of extended and classical nonlinear programming with twice continuously differentiable data

solutions to the master equations governing fractional vortices abstract by means of variational methods in this paper we establish sharp existence results for solutions of the master equations governing fractional multiple vortices in the doubly periodic situation the conditions for existence are both necessary and sufficient and give the upper bounds for the vortex numbers in terms of the size of the periodic cell domain in the planar situation there is no restriction on the vortex numbers in both situations the solutions are uniquely determined by the prescribed locations and the local winding numbers of the vortices

on the positivity of the effective action in a theory of random surfaces it is shown that the functional s eta frac 1 24 pi int left frac 1 2 left nabla eta right 2 2 eta right d mu 0 defined onc functions on the two dimensional sphere satisfies the inequalitys 0 if is subject to the constraint int e eta 1 d mu 0 0 the minimums 0 is attained at the solutions of the euler lagrange equations the proof is based on a sharper version of moser trudinger s inequality due to aubin which holds under the additional constraint int e eta xd mu 0 0 this condition can always be satisfied by exploiting the invariance ofs under the conformal transformations ofs2 the result is relevant for a recently proposed formulation of a theory of random surfaces

a singular sphere covering inequality uniqueness and symmetry of solutions to singular liouville type equations we derive a singular version of the sphere covering inequality which was recently introduced in gui and moradifam invent math https doi org 10 1007 s00222 018 0820 2 2018 suitable for treating singular liouville type problems with superharmonic weights as an application we deduce new uniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains as well as new self contained proofs of previously known results such as the uniqueness of spherical convex polytopes first established in luo and tian proc am math soc 116 4 1119 1129 1992 furthermore we derive new symmetry results for the spherical onsager vortex equation

compact isospectral sets of surfaces on etudie des ensembles de surfaces qui sont isospectrales par rapport a l operateur de laplace beltrami

fully holderian stable minimum with respect to both tilt and parameter perturbations when the objective function undergoes both a tilt perturbation and a general parameter perturbation this paper considers the notions of a fully stable holder minimizer a uniform holder growth con

second order characterizations of tilt stability with applications to nonlinear programming the paper is devoted to the study of tilt stable local minimizers of general optimization problems in finite dimensional spaces and its applications to classical nonlinear programs with twice continuously differentiable data the importance of tilt stability has been well recognized from both theoretical and numerical aspects of optimization and this notion has been extensively studied in the literature based on advanced tools of second order variational analysis and generalized differentiation we develop a new approach to tilt stability which allows us to derive not only qualitative but also quantitative characterizations of tilt stable minimizers with calculating the corresponding moduli the implementation of this approach and general results in the classical framework of nonlinear programming provides complete characterizations of tilt stable minimizers under new second order qualification and optimality conditions

full lipschitzian and holderian stability in optimization with applications to mathematical programming and optimal control the paper concerns a systematic study of full stability in general optimization models including its conventional lipschitzian version as well as the new holderian one we derive various characteri

some properties of fuzzy groups on definit des sous groupes flous normaux des sous groupes flous normaux de niveau et leurs homomorphismes et on donne certaines de leurs proprietes

some characterizations of fuzzy subgroups abstract we show that if a fuzzy subgroup can be written as the intersection of all its minimal fuzzy p subgroups or the intersection of all its minimal fuzzy p subgroups then some properties of the fuzzy subgroup characterize the properties of all its minimal fuzzy p subgroups or all its minimal fuzzy p subgroups and investigate interrelationships among some properties of fuzzy subgroups

fuzzy normal subgroups and fuzzy cosets abstract we introduce here the notion of 1 i a fuzzy normal subgroup and 2 ii a fuzzy coset we prove that for a group g a fuzzy subgroup is fuzzy normal if and only if it is constant on the conjugate classes of g further we show that the level subgroups of a fuzzy normal subgroup are all normal we obtain several results which are analogs of some basic theorems of group theory in particular a fuzzy version of lagrange s theorem is established

extremals for moser inequalities this paper shows that extremals exist for four moser inequalities involving exponential integrals the results are for sobolev spaces defined on the spheres s2 and s3 on the disk d2 and on rp2 the method that carleson and chang used for the n ball with boundary value zero can be modified and applied to functions with mean value zero on these domains this approach also provides an elementary proof of the moser inequality for sn n 2 which is a special case of a result of fontana

feasible methods for nonconvex nonsmooth problems with applications in green communications we propose a general feasible method for nonsmooth nonconvex constrained optimization problems the algorithm is based on the inexact solution of a sequence of strongly convex optimization subproblems followed by a step size procedure key features of the scheme are i it preserves feasibility of the iterates for nonconvex problems with nonconvex constraints ii it can handle nonsmooth problems and iii it naturally leads to parallel distributed implementations we illustrate the application of the method to an open problem in green communications whereby the energy consumption in mimo multiuser interference networks is minimized subject to nonconvex quality of service constraints

tilt stability uniform quadratic growth and strong metric regularity of the subdifferential we prove that uniform second order growth tilt stability and strong metric regularity of the subdifferential three notions that have appeared in entirely different settings are all essentially equivalent for any lower semicontinuous extended real valued function

sharp moser trudinger inequalities for the laplacian without boundary conditions we derive a sharp moser trudinger inequality for the borderline sobolev imbedding of w2 n 2 bn into the exponential class where bn is the unit ball of rn the corresponding sharp results for the spaces w0d n d are well known for general domains and are due to moser and adams when the zero boundary condition is removed the only known results are for d 1 and are due to chang yang cianchi and leckband the proof of our result is based on a new integral representation formula for the canonical solution of the poisson equation on the ball that is the unique solution of the equation u f which is orthogonal to the harmonic functions on the ball the main technical difficulty of the paper is to establish an asymptotically sharp growth estimate for the kernel of such representation expressed in terms of its distribution function

the scalar curvature flow on s n perturbation theorem revisited for the problem of finding a geometry on s n for n 3 with a prescribed scalar curvature there is a well known result which is called the perturbation theorem it is due to chang and yang duke math j 64 27 69 1991 their key assumption is that the candidate f for the prescribed scalar curvature is sufficiently near the scalar curvature of the standard metric in the sup norm it is important to know how large that difference in sup norm can possibly be here we consider prescribing scalar curvature problem using the scalar curvature flow

on generalized characteristic fuzzy subgroups of a finite group abstract in 1991 asaad 2 suggested the following two questions i what can be said about the structure of a fuzzy subgroup a of a finite group g when information is known about the structure of g ii what can be said about the structure of group g when information is known about the structure of a fuzzy subgroup a of g the purpose of this paper is to continue the investigation of the above questions

full stability of locally optimal solutions in second order cone programs the paper presents complete characterizations of lipschitzian full stability of locally optimal solutions to second order cone programs socps expressed entirely in terms of their initial data these characterizations are obtained via appropriate versions of the quadratic growth and strong second order sufficient conditions under the corresponding constraint qualifications we also establish close relationships between full stability of local minimizers for socps and strong regularity of the associated generalized equations at nondegenerate points our approach is mainly based on advanced tools of second order variational analysis and generalized differentiation

partial smoothness tilt stability and generalized hessians we compare two recent variational analytic approaches to second order conditions and sensitivity analysis for nonsmooth optimization we describe a broad setting where computing the generalized hessian of mordukhovich is easy in this setting the idea of tilt stability introduced by poliquin and rockafellar is equivalent to a classical smooth second order condition

application of an adams type inequality to a two chemical substances chemotaxis system abstract this paper deals with positive solutions of the fully parabolic system u t u u v in 0 1 v t v v w in 0 2 w t w w u in 0 under homogeneous neumann boundary conditions or mixed boundary conditions no flux and dirichlet conditions in a smooth bounded domain r n n 4 with positive parameters 1 2 0 and nonnegative smooth initial data u 0 v 0 w 0 in the lower dimensional case n 3 it is proved that for all reasonable initial data solutions of the system exist globally in time and remain bounded in the case n 4 it is shown that in the radially symmetric setting solutions to the neumann boundary value problem of the system exist globally in time and remain bounded if u 0 l 1 8 2 as to the mixed boundary value problem we will establish global existence and boundedness of solutions if u 0 l 1 8 2 without radial symmetry the key ingredients are a lyapunov functional and an adams type inequality a lyapunov functional of the above problems will be constructed and the constant 8 2 is deduced from the critical constant in the adams type inequality this result is regarded as a generalization of the well known 8 problem in the keller segel system to higher dimensions

local monotonicity and full stability for parametric variational systems this paper introduces and characterizes new notions of lipschitzian and holderian full stability of solutions to general parametric variational systems defined via partial subdifferential of prox regular functions acting in finite dimensional and hilbert spaces these notions which postulate certain quantitative properties of single valued localizations of solution maps are closely related to local strong maximal monotonicity of associated set valued mappings based on advanced tools of variational analysis and generalized differentiation we derive verifiable characterizations of the local strong maximal monotonicity and full stability notions under consideration via some positive definiteness conditions involving second order constructions of variational analysis

non degeneracy and uniqueness of solutions to singular mean field equations on bounded domains abstract the aim of this paper is to complete the program initiated in 51 23 and then carried out by several authors concerning non degeneracy and uniqueness of solutions to mean field equations in particular we consider mean field equations with general singular data on non smooth domains the argument is based on the alexandrov bol inequality and on the eigenvalues analysis of linearized singular liouville type problems

conformal deformation of metrics on s 2 soit d un domaine lisse de s 2 et k une fonction lisse definie sur d alors pour que le probleme aux limites de neumann u k exp 2u 1 sur d u n 0 sur d ait une solution u on a a quand aire d max x d k x 0

prox regular functions in hilbert spaces this paper studies the prox regularity concept for functions in the general context of hilbert space in particular a subdifferential characterization is established as well as several other properties it is also shown that the moreau envelopes of such functions are continuously differentiable

fuzzy subgroups having the property abstract we introduce the notion of the property for a fuzzy group and characterize all finite cyclic groups in terms of this notion this property concerns the transition of the order of an element of a group from the classical to the fuzzy setting

a new approach to sharp moser trudinger and adams type inequalities a rearrangement free argument abstract the main purpose of this paper is two fold on the one hand we will develop a new approach to establish sharp singular moser trudinger and adams type inequalities in unbounded domains of euclidean spaces without using the standard symmetrization on the other hand we will prove the sharp singular adams type inequality on high order sobolev spaces w m n m r n of arbitrary integer order m theorem 1 1 which improves the results of ruf and sani 2013 48 where sharp adams inequalities were established for even m and those of the authors lam and lu 2012 28 29 for odd m but with different and more restricted norms we first establish the sharp local singular adams inequality on domains in r n of finite measure theorem 1 4 we take a perspective that any function in the high order sobolev spaces w m n m r n can be represented as a bessel potential thus we can fully use the tools from harmonic analysis and the kernel properties of the polyharmonic operators i m 2 once we have established this sharp local adams inequality then we can adapt the rearrangement free method we will develop in this paper to derive a global sharp adams inequality from a local one our argument substantially simplifies those in ruf and sani 2013 48 and lam and lu 2012 28 29 and avoids the use of rather deep and complicated comparison principle of solutions to polyharmonic operators used in ruf and sani 2013 48 lam and lu 2012 28 29 moreover our theorem holds on sobolev spaces w n r n of any positive fractional order n theorems 1 6 and 1 7 as consequences we can also establish sharp adams inequalities with less restrictions on the sobolev norms see theorems 1 2 1 3 and 1 5 our approach is surprisingly simple and general and can be easily applied to scenarios such as riemannian and sub riemannian manifolds where symmetrization argument does not work see e g on the heisenberg group lam and lu 2012 30

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

holder weak sharp minimizers and holder tilt stability abstract in this paper we introduce and study the notions of holder weak sharp minimizers stable holder weak sharp minimizers and holder tilt stable weak minimizers for a proper lower semicontinuous function f on a banach space in terms of the holder metric subregularity regularity of f we consider optimality conditions for holder weak sharp minimizers and stable holder weak sharp minimizers we prove that x is a stable holder weak sharp minimizer resp a holder tilt stable weak minimizer of f if and only if it is a stable holder sharp minimizer resp a holder tilt stable minimizer of f

tl subgroups having the property abstract we introduce the notion of a tl subgroup having the property that is a generalization of the notion of a generalized characteristic tl subgroup as abou zaid s notion was generalized in kim 1994 5 and investigate interrelationship between the notion of a tl subgroup having the property and the notion of a generalized characteristic tl subgroup

second order variational analysis and characterizations of tilt stable optimal solutions in infinite dimensional spaces abstract the paper is devoted to developing second order tools of variational analysis and their applications to characterizing tilt stable local minimizers of constrained optimization problems infinite dimensional spaces with many results new also in finite dimensional settings the importance of tilt stability has been well recognized from both theoretical and numerical aspects of optimization based on second order generalized differentiation we obtain qualitative and quantitative characterizations of tilt stability in general frameworks of constrained optimization and establish its relationships with strong metric regularity of subgradient mappings and uniform second order growth the results obtained are applied to deriving new necessary and sufficient conditions for tilt stable minimizers in problems of nonlinear programming with twice continuously differentiable data in hilbert spaces

extremal metrics of zeta function determinants on 4 manifolds in conformal geometry the sobolev inequality at a critical exponent has received much attention in particular the determination of the best constants has played a crucial role in the yamabe problem in dimension two the analogous problem deals with the moser trudinger inequality on a compact riemann surface m2 there exists a constant c c m so that f e47w2 c m if f ivw12 1 and f w 0 the connection of this inequality with geometry comes through the zeta functional determinant of the laplacian as defined by ray singer for a riemannian metric g let 0 a1 a2 be the spectrum

on a sharp sobolev type inequality on two dimensional compact manifolds motivated by the asymptotic analysis of double vortex condensates in the chern simons higgs theory we construct a suitable minimizing sequence for a sharp sobolev inequality a la moser for two dimensional compact manifolds as a consequence we first obtain a direct proof of the sharp character of such an inequality secondly and more interestingly we use such minimizing sequence to show that for the flat torus the corresponding extremal problem attains its infimum

fuzzy orders relative to fuzzy subgroups abstract in this paper we introduce the notion of fuzzy orders of the elements of a group and show that most of the basic properties of orders of elements in the group theory are valid in the theory of fuzzy groups where ordinary orders of elements are replaced by fuzzy orders of elements reports of further development of the subject will follow in subsequent essays

on the best constant in the moser onofri aubin inequality let s2 be the 2 dimensional unit sphere and let j denote the nonlinear functional on the sobolev space h1 s2 defined by j alpha u frac alpha 16 pi int s 2 nabla u 2 d mu 0 frac 1 4 pi int s 2 u d mu 0 rm ln int s 2 e u frac d mu 0 4 pi where d mu 0 sin theta d theta wedge d phi onofri had established that j is non negative on h1 s2 provided 1 in this note we show that if j is restricted to those u in h 1 s 2 that satisfies the aubin condition int s 2 e u x j d mu 0 0 quad rm for all 1 leq j leq 3 then the same inequality continues to hold i e j u 0 whenever alpha geq frac 2 3 epsilon 0 for some epsilon 0 0 the question of chang yang on whether this remains true for all alpha geq frac 1 2 remains open

variational analysis and full stability of optimal solutions to constrained and minimax problems abstract the main goal of this paper is to develop applications of advanced tools of first order and second order variational analysis and generalized differentiation to the fundamental notion of full stability of local minimizers of general classes of constrained optimization and minimax problems in particular we derive second order characterizations of full stability and investigate its relationships with other notions of stability for parameterized conic programs and minimax problems furthermore the developed variational approach allows us to largely unify and provide new self contained proofs of some quite recent results in this direction for problems of constrained optimization with c 2 smooth data

isomorphic fuzzy groups abstract we introduce an acceptable definition of isomorphic fuzzy groups which fulfils the basic requirements the fuzzy versions of the first and the second isomorphism theorems become immediate consequences the idea is applied to the sup min product of two fuzzy subgroups of a group as an indispensable tool the characterization of a fuzzy normal subgroup of a group by the so called open level subgroups is brought forth an isomorphism between the cartesian product and the sup min product of two fuzzy subgroups of a group is demonstrated

a quantization property for blow up solutions of singular liouville type equations abstract motivated by the study of self dual vortices in gauge field theory cf vortices and monopoles birkhauser boston 1980 solitons in field theory and nonlinear analysis monographs in mathematics springer new york 2001 we consider a concentrating solution sequence u k satisfying u k z 2 k v k z e u k in b 1 z 2 k v k z e u k z 0 weakly in the sense of measures in b 1 where i k r k ii v k c 0 1 b 1 0 a v k b and v k a in b 1 we prove that necessarily 8 n 8 1 8 z the result is sharp as shown by explicit examples and should be compared with that obtained by li shafrir ind univ math j 43 4 1994 1255 concerning the situation where k 0 k n

a study of tilt stable optimality and sufficient conditions abstract recent results by eberhard et al 2006 4 and eberhard and wenczel 2009 3 on the interaction of single and double envelope operations of nonsmooth functions and their interaction with second order derivations have been used to study tilt stability of local minima this continues the study begun by poliquin and rockafellar 1998 1 but now armed with new tools we are able to make some new observations we observe that tilt stability entails a local density within the graph of the proximal subderivative of strict local minima order two of the tilted function indeed it also entails the strict local minimality order two of the tilt stable local minimum itself for prox regular subdifferentially continuous functions this density property characterises tilt stability

adams inequalities for bi laplacian and extremal functions in dimension four let be a smooth oriented bounded domain be the sobolev space and be the first eigenvalue of the bi laplacian operator 2 then for any a 0a o we have and the above supremum is infinity when a o this strengthens adams inequality in dimension 4 d adams a sharp inequality of j moser for high order derivatives ann of math 128 1988 365 398 where he proved the above inequality holds for a 0 moreover we prove that for sufficiently small a an extremal function for the above inequality exists as a special case of our results we thus show that there exists with such that this establishes the existence of an extremal function of the original adams inequality in dimension 4

generalized differentiation of piecewise linear functions in second order variational analysis abstract the paper is devoted to a comprehensive second order study of a remarkable class of convex extended real valued functions that is highly important in many aspects of nonlinear and variational analysis specifically those related to optimization and stability this class consists of lower semicontinuous functions with possibly infinite values on finite dimensional spaces which are labeled as piecewise linear ones and can be equivalently described via the convexity of their epigraphs in this paper we calculate the second order subdifferentials generalized hessians of arbitrary convex piecewise linear functions together with the corresponding geometric objects entirely in terms of their initial data the obtained formulas allow us in particular to justify a new exact equality type second order sum rule for such functions in the general nonsmooth setting

topological degree for a mean field equation on riemann surfaces we consider the following mean field equations 0 1 where m is a compact riemann surface with volume 1 h is a positive continuous function on m is a real number 0 2 and where is a bounded smooth domain h is a c1 positive function on and based on our previous analytic work 14 we prove among other things that the degree counting formula for 0 1 is given by for 8m 8 m 1 2003 wiley periodicals inc

directional derivatives of the solution of a parametric nonlinear program consider a parametric nonlinear optimization problem subject to equality and inequality constraints conditions under which a locally optimal solution exists and depends in a continuous way on the parameter are well known we show under the additional assumption of constant rank of the active constraint gradients that the optimal solution is actually piecewise smooth hence b differentiable we show for the first time to our knowledge a practical application of quadratic programming to calculate the directional derivative in the case when the optimal multipliers are not unique

remarks on prescribing gauss curvature we study the nonlinear partial differential equation for the problem of prescribing gauss curvature k on s 2 we give an example of a rotationally symmetric k for which the kazdan warner obstruction is satisfied but the equation has no rotationally symmetric solution on the other hand we give a simple sufficient condition for solvability of the equation when k is rotationally symmetric finally we give a sufficient condition for solvability when k is not necessarily rotationally symmetric

groups and fuzzy subgroups abstract the purpose of this paper is to deal with the following two problems 1 what can be said about the structure of a fuzzy subgroup a of a group g when information is known about the structure of g 2 what can be said about the structure of a group g when information is known about the structure of a fuzzy subgroup a of g