sharp estimates for solutions of multi bubbles in compact riemann surfaces in this paper we consider a sequence of multibubble solutions u k of the equation 0 1 0 u k k he uk f m he u k d 1 0 in m where h is a c 2 positive function in a compact riemann surface m and k is a constant satisfying lim k k 8m for some positive integer m 1 we prove among other things that k 8m 2 m m j 1h 1 p k j 0 log h p k j 8m 2k p k j k j e k j o e k j where p k j are centers of the bubbles of u k and k j are the local maxima of u k after adding a constant this yields a uniform bound of solutions as k converges to 8m from below provided that 0 log h p k j 8m 2k p k j 0 it generalizes a previous result due to ding jost li and wang 18 and nolasco and tarantello 31 which says that any sequence of minimizers u k is uniformly bounded if k 0 for any maximum point p of the sum of 2 log h and the regular part of the green function where k is the gaussian curvature of m the analytic work of this paper is the first step toward computing the topological degree of 0 1 which was initiated by li 24

existence of bubbling solutions for chern simons model on a torus we study the existence of bubbling solutions for the the following chern simons higgs equation delta u frac1 varepsilon 2 rm e u 1 rm e u 4 pi sum i 1 2k delta p i quad text in omega where is a torus if k 1 for any critical point q of the associated sum of the green functions we introduce a quantity d q see 1 11 below we show that for any non degenerate critical point q with d q 0 the above problem has a solution ue satisfying that e 0 ue blows up at q the calculations in this paper also show that if a sequence of solutions ue blows up at q as e 0 then q must be a critical point of the associated sum of the green functions and d q leqq 0 so the condition d q 0 is almost necessary to obtain our result we also construct solutions with k bubbles for k geqq 2

statistical mechanics of classical particles with logarithmic interactions the inhomogeneous mean field thermodynamic limit is constructed and evaluated for both the canonical thermodynamic functions and the states of systems of classical point particles with logarithmic interactions in two space dimensions the results apply to various physical models of translation invariant plasmas gravitating systems as well as to planar fluid vortex motion for attractive interactions a critical behavior occurs which can be classified as an extreme case of a second order phase transition to include in particular attractive interactions a new inequality for configurational integrals is derived from the arithmetic geometric mean inequality the method developed in this paper is easily seen to apply as well to systems with fairly general interactions in all space dimensions in addition it also provides us with a new proof of the trudinger moser inequality known from differential geometry in its sharp form

mean field equation of liouville type with singular data topological degree we consider the following mean field equation gv h ev mh ev 1 4 j 1n j qj 1 on m where m is a compact riemann surface with volume 1 h is a positive c1 function on m and and j are constants satisfying j 1 in this paper we derive the topological degree counting formula for noncritical values of we also give several applications of this formula including existence of the curvature 1 metric with conic singularities doubly periodic solutions of electroweak theory and a special case of self gravitating strings 2015 wiley periodicals inc

a diffeomorphism classification of 7 dimensional homogeneous einstein manifolds with mathrm su 3 times mathrm su 2 times mathrm u 1 symmetry on donne des exemples qui montrent que des espaces homogenes homeomorphes ne sont pas necessairement diffeomorphes

singular mean field equations on compact riemann surfaces for a general class of elliptic pde s in mean field form on compact riemann surfaces with exponential nonlinearity we address the question of the existence of solutions with concentrated nonlinear term which in view of the applications are physically of definite interest in the model we also include the possible presence of singular sources in the form of dirac masses which makes the problem more difficult to attack

inference of fuzzy relational tableaux from fuzzy exemplifications the paper describes a method for deriving fuzzy relational tableaux from fuzzy exemplifications in particular the work by thole zimmerman and zysno 4 is cited as they derive an operator for and using data derived from human subjects the method results in an operator or operators which is a maximal restriction of the operators implied by the data results showing the process are described and a value of and is derived using data presented in 4

an improved geometric inequality via vanishing moments with applications to singular liouville equations we consider a class of singular liouville equations on compact surfaces motivated by the study of electroweak and self dual chern simons theories the gaussian curvature prescription with conical singularities and onsager s description of turbulence we analyse the problem of existence variationally and show how the angular distribution of the conformal volume near the singularities may lead to improvements in the moser trudinger inequality and in turn to lower bounds on the euler lagrange functional we then discuss existence and non existence results

homogeneous einstein metrics and simplicial complexes on compact homogeneous spaces we investigate the hilbert action restricted to the space of homogeneous metrics of volume one based on a detailed understanding of the high energy levels of this action we assign to a compact homogeneous space a simplicial complex whose non contractibility is a sufficient condition for the existence of homogeneous einstein metrics

blow up phenomena for the liouville equation with a singular source of integer multiplicity abstract we are concerned with the existence of blowing up solutions to the following boundary value problem u a x e u 4 n 0 in u 0 on where is a smooth and bounded domain in r 2 such that 0 a x is a positive smooth function n is a positive integer and 0 is a small parameter here 0 defines the dirac measure with pole at 0 we find conditions on the function a and on the domain under which there exists a solution u blowing up at 0 and satisfying a x e u 8 n 1 as 0

the properties of fuzzy divisible groups abstract f i sidky and m atif mishref in 1990 introduced the concepts of divisible and pure fuzzy subgroups and proved some of their properties in this paper we further prove the following results if f g g is an isomorphism of groups and a is a divisible pure fuzzy subgroup of under min then f 1 a is a divisible pure fuzzy subgroup of g if b is a fuzzy1 subgroup of divisible subgroup h of g then there exists a subgroup h ofg and a fuzzy subgroup b of h such that g h h and a b b is a fuzzy subgroup of g if g is a divisible cyclic group then g is a finite group if a is a fuzzy subgroup of divisible cyclic group then a is trivial consider group g g1 g2 gm with gi a divisible cyclic subgroup of g i 1 2 m and suppose a is a divisible pure fuzzy subgroup of g under min then a is trivial

a special class of stationary flows for two dimensional euler equations a statistical mechanics description we consider the canonical gibbs measure associated to an vortex system in a bounded domain at inverse temperature widetilde beta and prove that in the limitn widetilde beta n n 1 where 8 here denotes the vorticity intensity of each vortex the one particle distribution function n nx x converges to a superposition of solutions of the following mean field equation left begin array 20 c varrho beta x frac e beta psi mathop smallint limits lambda e beta psi delta psi varrho beta in lambda psi partial lambda 0 end array right

bubbling solutions for the liouville equation with a singular source non simple blow up abstract we are concerned with the existence of blowing up solutions to the following boundary value problem u e u 4 n 0 in u 0 on where is a smooth and bounded domain in r 2 such that 0 n is a positive number close to an integer n n 1 from the right side 0 defines the dirac measure with pole at 0 and 0 is a small parameter we assume that is n 1 symmetric and the regular part of the green s function satisfies a nondegeneracy condition both assumptions are verified if is the unit ball and we find a solution which exhibits a non simple blow up profile as 0

new invariant einstein metrics on generalized flag manifolds a generalized flag manifold or a kahlerian c space is a homogeneous space g k whose isotropy subgroup k is the centralizer of a torus in g these spaces admit a finite number of kahler einstein metrics we present new non kahler einstein metrics for certain quotients of u n so 2n and g 2 we also examine the isometry question for these metrics

on a class of elliptic problems in r symmetry and uniqueness results in the plane r 2 we classify all solutions for an elliptic problem of liouville type involving a radial weight function as a consequence we clarify the origin of the non radially symmetric solutions for the given problem as established by chanillo and kiessling for a more general class of liouville type problems we show that rather than radial symmetry the solutions always inherit the invariance of the problem under inversion with respect to suitable circles this symmetry result is derived with the help of a shrinking sphere method

ricci iteration on homogeneous spaces the ricci iteration is a discrete analogue of the ricci flow we give the first study of the ricci iteration on a class of riemannian manifolds that are not kahler the ricci iteration in the non kahler setting exhibits new phenomena among them is the existence of so called ancient ricci iterations as we show these are closely related to ancient ricci flows and provide the first nontrivial examples of riemannian metrics to which the ricci operator can be applied infinitely many times in some of the cases we study these ancient ricci iterations emerge in the gromov hausdorff topology from a collapsed einstein metric and converge smoothly to a second einstein metric in the case of compact homogeneous spaces with maximal isotropy we prove a relative compactness result that excludes collapsing our work can also be viewed as proposing a dynamical criterion for detecting whether an ancient ricci flow exists on a given riemannian manifold as well as a method for predicting its limit

weighted barycentric sets and singular liouville equations on compact surfaces given a closed surface we prove a general existence result for some elliptic pde with exponential nonlinearities and negative dirac deltas extending a theory recently obtained for the regular case this is done by global methods since the associated euler functional might be unbounded from below we define a new model space generalizing the so called space of formal barycenters and characterizing up to homotopy equivalence its low sublevels as a result the analytic problem is reduced to a topological one concerning the contractibility of this model space to this aim we prove a new functional inequality in the spirit of chen and li 1991 11 and then employ a min max scheme based on conical construction jointly with the blow up analysis in bartolucci and montefusco 2007 4 after bartolucci and tarantello 2002 brezis and merle 1991 and this study is motivated by abelian chern simons theory in self dual regime or from the problem of prescribing the gaussian curvature with conical singularities generalizing a problem raised in kazdan and warner 1974 24

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

existence and non existence of homogeneous einstein metrics on demontre un theoreme d existence general pour des metriques d einstein homogenes et on presente des espaces homogenes compacts simplement connexes qui ne portent pas une metrique d einstein homogene

kahler einstein metrics with positive scalar curvature in this paper we prove that the existence of kahler einstein metrics implies the stability of the underlying kahler manifold in a suitable sense in particular this disproves a long standing conjecture that a compact kahler manifold admits kahler einstein metrics if it has positive first chern class and no nontrivial holomorphic vector fields we will also establish an analytic criterion for the existence of kahler einstein metrics our arguments also yield that the analytic criterion is satisfied on stable kahler manifolds provided that the partial c 0 estimate posed in t6 is true

the alekseevskii conjecture in low dimensions the long standing alekseevskii conjecture states that a connected homogeneous einstein space g k of negative scalar curvature must be diffeomorphic to mathbb r n this was known to be true only in dimensions up to 5 and in dimension 6 for non semisimple g in this work we prove that this is also the case in dimensions up to 10 when g is not semisimple for arbitrary g besides 5 possible exceptions we show that the conjecture holds up to dimension 8

divisible and pure fuzzy subgroups abstract as a continuation to the study of fuzzy subgroups which was given by a rosenfeld s sessa p s das h sherwood and others we introduce the concepts of divisible and pure fuzzy subgroups and prove some of their properties

sign changing blow up solutions for henon type elliptic equations with exponential nonlinearity abstract we study the existence of sign changing solutions with multiple concentration to the following boundary value problem u e 2 x 2 e u e u in u 0 on where 0 is a smooth bounded domain in r 2 containing the origin e 0 is a small parameter in particular we prove that if 1 then a nodal solution exists with a number of mixed positive and negative blow up points up to 4

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

divisible tl subgroups and pure tl subgroups abstract as a continuation of the study of divisible and pure fuzzy subgroups see sidky and mishref 1990 we introduce the concepts of divisible tl subgroups and pure tl subgroups of an additive group where t is an arbitrary infinitely distributive t norm on a given complete brouwerian lattice l and show some of their properties

products of fuzzy subgroups fuzzy subgroups are different from ordinary subgroups in that one cannot tell with certainty which group elements belong and which do not of course this requires an appropriate modification of the closure property which takes the form of an inequality in this paper a study of products of fuzzy subgroups is begun some results are expected the product of fuzzy subgroups is a fuzzy subgroup products with the same but permuted factors are isomorphic and there is a natural isomorphism from any factor into the product functions called t norms are used to construct products of fuzzy subgroups different t norms yielding different products since a fuzzy subgroup must satisfy an inequality generalizing closure and the strength of that inequality is important some results relate the t norm used in forming the product to the strength of the ensuing inequality when can complicated fuzzy subgroups be expressed as products of simple ones this question is addressed for fuzzy subgroups of finite abelian groups

two dimensional euler flows with concentrated vorticities for a planar model of euler flows proposed by tur and yanovsky 2004 we construct a family of velocity fields w e for a fluid in a bounded region with concentrated vorticities e for e 0 small more precisely given a positive integer and a sufficiently small complex number a we find a family of stream functions e which solve the liouville equation with dirac mass source e e 2 e e 4 pa e in e 0 on for a suitable point p p a e the vorticities e e concentrate in the sense that 1 e 4 pa e 8 1 j 1 pa e a j 0 as e 0 j 1 where the satellites a 1 a 1 denote the complex 1 roots of a the point p a e lies close to a zero point of a vector field explicitly built upon derivatives of order 1 of the regular part of green s function of the domain