smooth embeddings of sl 2 and pgl 2 an embedding of an algebraic homogeneous space g h is an algebraic normal variety with an action of g having an open dense orbit isomorphic to g h there is a beautiful theory for the embeddings of tori see for example kkms danl or oda in this theory the embeddings are classified by fans of rational convex cones by studying the combinatorics of the fans one can determine certain geometric properties of the associated embeddings for example one can check which embeddings are smooth complete or projective recently luna and vust developed a method to classify embeddings of homogeneous spaces g h where g is a connected algebraic reductive group lv in the present paper we analyse more closely the cases of embeddings of x 2 and pgl 2 each embedding is represented by a diagram containing combinatorical information about the local rings of orbits as in the case of torus embeddings one would like to be able to determine geometric properties from these diagrams in this article we find the conditions that the diagrams must satisfy to represent smooth embeddings the idea of the proof is as follows an embedding is smooth if and only if the local rings of all the orbits are regular the embedding is a three dimensional normal variety so all orbits of dimension two are smooth it can be shown that the fixed points are never smooth so one is left to treat the one dimensional orbits which are isomorphic to the projective line a result from blv states that a point of a one dimensional orbit has an affme neighborhood which is the product of the affrne line and a certain two dimensional variety the orbit projects down to a point in the two dimensional variety thus the problem is reduced to checking the regularity of certain local rings of points in twodimensional varieties in each case either one can find explicitly two

calculus for interval valued functions using generalized hukuhara derivative and applications this paper is devoted to studying differential calculus for interval valued functions by using the generalized hukuhara differentiability which is the most general concept of differentiability for interval valued functions conditions examples and counterexamples for limit continuity integrability and differentiability are given special emphasis is set to the class f t c g t where c is an interval and g is a real function of a real variable here the emphasis is placed on the fact that f and g do not necessarily share their properties underlying the extra care that must be taken into account when dealing with interval valued functions two applications of the obtained results are presented the first one determines a delta method for interval valued random elements in the second application a new procedure to obtain solutions to an interval differential equation is introduced our results are relevant to fuzzy set theory because the usual fuzzy arithmetic extension functions and mathematical analysis are done on a cuts which are intervals

invariant functions on symplectic representations abstract we study invariants on symplectic representations of a connected reductive group our main result is that the invariant moment map is equidimensional we deduce that the categorical quotient is a fibration over an affine space with rational generic fibers of particular interest are representations where these fibers are points we show that they are cofree our main tool is a symplectic version of the local structure theorem

a restriction theorem for modules having a spherical submodule we introduce the following notion a finite dimensional represen tation v of a complex reductive algebraic group g is called spherical of rank one if the generic stabilizer m is reductive the pair g m is spherical and dim vm 1 let u be another finite dimensional representation of g we denote by s u s u g the ring of polynomial functions on u the ring of g invariant polynomial functions on u we characterize the image of s u v g under the restriction map into s u vm as the w ng m m invariants in the rees ring associated to an ascending filtration of s u m furthermore under some additional hypothesis we give an isomorphism be tween the graded ring associated to that filtration and s u p where p is the stabilizer of an unstable point whose 7 orbit has maximal dimension

a new approach to linear interval differential equations as a first step toward solving fuzzy differential abstract this study uses a concept of interval differentiability which was recently introduced to formulate interval initial value problems involving linear interval differential equations differently from the approaches that use the gh differentiability this study does not make use of a criterion of choice for switching points in order to obtain solutions for such problems the method herein presented provides solutions in a simple straightforward and computationally tractable way moreover these solutions are intuitive because they coincide with the solutions given by a differential inclusion method the efficiency and practicality of our approach are illustrated through some examples that have appeared in other articles

a connectedness property of algebraic moment maps let k be a connected compact lie group and m a hamiltonian k manifold i e a symplectic k manifold equipped with a moment map m k liek a theorem of kirwan implicitly in ki asserts if m is connected and compact then the level sets of are connected the purpose of this note is to prove such a statement in the category of algebraic varieties first we reformulate kirwan s theorem consider the map m k k which is the composition of with the quotient map for a point x k let h kx be its isotropy group and y kx k k its orbit then the fiber y is isomorphic to the fiber product k x thus since both k h and h are connected the connectedness of the fibers of is equivalent to the connectedness of the fibers of this formulation is more suitable for the algebraic category let g be a connected reductive group everything over c and z a hamiltonian gvariety with moment map z g lieg let g g specc g be the categorical quotient and let z g g be the composition of with the quotient map since the latter is not an orbit map the connection between fibers of and is much more loose than in the differential category and we concentrate on from now on the morphism is still not the right map since sometimes not even its generic fibers are connected an example is the action of g sl2 c on z c 2 c this is the cotangent bundle of c then g g a and u u in particular the generic fiber of has two connected components this is remedied by looking at the map u u instead then is the composition of with the finite map a a z 7 z the latter being responsible for the disconnected fibers this construction can be generalized as follows the morphism induces an homomorphism of algebras

on the structure of quasi stationary competing particle systems we study point processes on the real line whose configurations x are locally finite have a maximum and evolve through increments which are functions of correlated gaussian variables the correlations are intrinsic to the points and quantified by a matrix q q ij i j a probability measure on the pair x q is said to be quasi stationary if the joint law of the gaps of x and of q is invariant under the evolution a known class of universally quasi stationary processes is given by the ruelle probability cascades rpc which are based on hierarchically nested poisson dirichlet processes it was conjectured that up to some natural superpositions these processes exhausted the class of laws which are robustly quasi stationary the main result of this work is a proof of this conjecture for the case where q ij assume only a finite number of values the result is of relevance for mean field spin glass models where the evolution corresponds to the cavity dynamics and where the hierarchical organization of the gibbs measure was first proposed as an ansatz

transportation cost inequalities on path and loop groups let g be a connected lie group with the lie algebra g the action of cameron martin space h g on the path space pe g introduced by l gross illinois j math 36 1992 447 is free using this fact we define the h distance on pe g which enables us to establish a transportation cost inequality on pe g this method will be generalized to the path space over the loop group le g so that we obtain a transportation cost inequality for heat measures on le g

local rigidity of quasi regular varieties for a g variety x with an open orbit we define its boundary x as the complement of the open orbit the action sheaf s x is the subsheaf of the tangent sheaf made of vector fields tangent to x we prove for a large family of smooth spherical varieties the vanishing of the cohomology groups h i x s x for i 0 extending results of bien and brion compos math 104 1 26 1996 we apply these results to study the local rigidity of the smooth projective varieties with picard number one classified in pasquier math ann in press

uber bewertungen welche unter einer reduktiven gruppe invariant sind letg be a reductive group defined over an algebraically closed fieldk and letx be ag variety in this paper we studyg invariant valuationsv of the fieldk of rational functions onx these objects are fundamental for the theory of equivariant completions ofx letb be a borel subgroup andu the unipotent radical ofb it is proved thatv is uniquely determined by its restriction toku then we study the set of invariant valuations having some fixed restrictionv0 tokb ifv0 is geometric i e induced by a prime divisor then this set is a polyhedron in some vector space in characteristic zero we prove that this polyhedron is a simplicial cone and in fact the fundamental domain of finite reflection groupwx thus the classification of invariant valuations is almost reduced to the classification of valuations ofkb

on the coordinate ring of spherical conjugacy classes let g be a simple algebraic group over an algebraically closed field k of characteristic zero and mathcal o be a spherical conjugacy class of g we determine the decomposition of the coordinate ring k mathcal o of mathcal o into simple g modules

characterization of invariant measures at the leading edge for competing particle systems we study systems of particles on a line which have a maximum are locally finite and evolve with independent increments quasi stationary states are defined as probability measures on the algebra generated by the gap variables for which joint distribution of gaps between particles is invariant under the time evolution examples are provided by poisson processes with densities of the form p d x e sx s dx with s 0 and linear superpositions of such measures we show that conversely any quasi stationary state for the independent dynamics with an exponentially bounded integrated density of particles corresponds to a superposition of poisson processes with densities p dx e sx s dx with s 0 restricted to the relevant algebra among the systems for which this question is of some relevance are spin glass models of statistical mechanics where the point process represents the collection of the free energies of distinct pure states the time evolution corresponds to the addition of a spin variable and the poisson measures described above correspond to the so called rem states

toric varieties and spherical embeddings over an arbitrary field abstract we are interested in two classes of varieties with group action namely toric varieties and spherical embeddings they are classified by combinatorial objects called fans in the toric setting and colored fans in the spherical setting we characterize those combinatorial objects corresponding to varieties defined over an arbitrary field k then we provide some situations where toric varieties over k are classified by galois stable fans and spherical embeddings over k by galois stable colored fans moreover we construct an example of a smooth toric variety under a 3 dimensional nonsplit torus over k whose fan is galois stable but which admits no k form in the spherical setting we offer an example of a spherical homogeneous space x 0 over r of rank 2 under the action of su 2 1 and a smooth embedding of x 0 whose fan is galois stable but which admits no r form

bounds on eigenvalues of real and complex interval matrices we present a cheap and tight formula for bounding real and imaginary parts of eigenvalues of real or complex interval matrices it outperforms the classical formulae not only for the complex case but also for the real case in particular it generalizes and improves the results by rohn 1998 5 and hertz 2009 19 the main idea behind is to reduce the problem to enclosing eigenvalues of symmetric interval matrices for which diverse methods can be utilized the result helps in analysing stability of uncertain dynamical systems since the formula gives sufficient conditions for testing schur and hurwitz stability of interval matrices it may also serve as a starting point for some iteration methods

random overlap structures properties and applications to spin glasses random overlap structures rost s are random elements on the space of probability measures on the unit ball of a hilbert space where two measures are identified if they differ by an isometry in spin glasses they arise as natural limits of gibbs measures under the appropriate algebra of functions we prove that the so called cavity mapping on the space of rost s is continuous leading to a proof of the stochastic stability conjecture for the limiting gibbs measures of a large class of spin glass models similar arguments yield the proofs of a number of other properties of rost s that may be useful in future attempts at proving the ultrametricity conjecture lastly assuming that the ultrametricity conjecture holds the setup yields a constructive proof of the parisi formula for the free energy of the sherrington kirkpatrick model by making rigorous a heuristic of aizenman sims and starr

bounding bar d distance by informational divergence a method to prove measure concentration there is a simple inequality by pinsker between variational distance and informational divergence of probability measures defined on arbitrary probability spaces we shall consider probability measures on sequences taken from countable alphabets and derive from pinsker s inequality bounds on the bar d distance by informational divergence such bounds can be used to prove the concentration of measure phenomenon for some nonproduct distributions

convex duality and the skorokhod problem ii in this paper we consider skorokhod problems on polyhedral domains with a constant and possibly oblique constraint direction specified on each face of the domain and with a corresponding cone of constraint directions at the intersection of faces in part one of this paper we used convex duality to develop new methods for the construction of solutions to such skorokhod problems and for proving lipschitz continuity of the associated skorokhod maps the main alternative approach to skorokhod problems of this type is the reflection mapping technique introduced by harrison and reiman 8 in this part of the paper we apply the theory developed in part one to show that the reflection mapping technique of 8 is restricted to a slight generalization of the class of problems originally considered in 8 we further illustrate the power of the duality approach by applying it to two other classes of skorokhod problems those with normal directions of constraint and a new class that arises from a model of processor sharing in communication networks in particular we prove existence of solutions to and lipschitz continuity of the skorokhod maps associated with each of these skorokhod problems

correspondence analysis with fuzzy data the fuzzy eigenvalue problem this paper constitutes a first step towards an extension of correspondence analysis with fuzzy data fca at this stage our main objective is to lay down the algebraic foundations for this fuzzy extension of the usual correspondence analysis a two step method is introduced to convert the fuzzy eigenvalue problem to an ordinary one we consider a fuzzy matrix as the set of its cuts each such cut is an interval valued matrix viewed as a line segment in the matrix space in this way line segments of cut matrices are transformed into intervals of eigenvalues therefore the two step method is essentially a reduction of the fuzzy eigenvalue problem to an ordinary one we illustrate the fca fuzzy eigenvalue problem with a simple numerical example we hope upon the completion of this project in near future to be able to supply the necessary tools for the end user of the correspondence analysis with fuzzy data

concentration for multidimensional diffusions and their boundary local times we prove that probability laws of certain multidimensional semimartingales which includes time inhomogenous diffusions under suitable assumptions satisfy quadratic transportation cost inequality under the uniform metric from this we derive concentration properties of lipschitz functions of process paths that depend on the entire history in particular we estimate concentration of boundary local time of reflected brownian motions on a polyhedral domain we work out explicit applications of consequences of measure concentration for the case of brownian motion with rank based drifts

a remark on the infinite volume gibbs measures of spin glasses in this note we point out that infinite volume gibbs measures of spin glass models on the hypercube can be identified as random probability measures on the unit ball of a hilbert space this simple observation follows from a result of dovbysh and sudakov on weakly exchangeable random matrices limiting gibbs measures can then be studied as single well defined objects this approach naturally extends the space of random overlap structures as defined by aizenman et al we discuss the ruelle probability cascades and the stochastic stability within this framework as an application we use an idea of parisi and talagrand to prove that if a sequence of finite volume gibbs measures satisfies the ghirlanda guerra identities then the infinite volume measure must be singular as a measure on a hilbert space

transportation cost information inequalities and applications to random dynamical systems and diffusions we first give a characterization of the l1 transportation cost information inequality on a metric space and next find some appropriate sufficient condition to transportation cost information inequalities for dependent sequences applications to random dynamical systems and diffusions are studied

eigenspace stability conditions in mean field replica theories i consider the eigenspace form of the saddle point conditions that apply for mean field replica models that are characterized by a single overlap matrix order parameter these allow for a characterization of the full set of saddle point solutions further they motivate consideration of a particular class of solutions which i call the principally commuting set these are the solutions that simultaneously satisfy the replica space rotational stability constraints for all physical models i show that this set satisfies the replica equivalence symmetry as well as other more stringent conditions and that it also contains the set of parisi type ultrametric solutions as a subset

information inequalities and concentration of measure we derive inequalities of the form p q h p r h q r which hold for every choice of probability measures p q r where h p r denotes the relative entropy of p with respect to r and p q stands for a coupling type distance between p and q using the chain rule for relative entropies and then specializing to q with a given support we recover some of talagrand s concentration of measure inequalities for product spaces

construction of pure states in mean field models for spin glasses if a mean field model for spin glasses is generic in the sense that it satisfies the extended ghirlanda guerra identities and if the law of the overlaps has a point mass at the largest point q of its support we prove that one can decompose the configuration space into a sequence of sets ak such that generically the overlap of two configurations is equal to q if and only if they belong to the same set ak for the study of the overlaps each set ak can be replaced by a single point combining this with a recent result of panchenko a connection between ghirlanda guerra identities and ultrametricity ann probab 2008 to appear this proves that if the overlaps take only finitely many values ultrametricity occurs we give an elementary self contained proof of this result based on simple inequalities and an averaging argument

quotients of some prehomogeneous vector spaces abstract we give quotients v g spec c v g for some prehomogeneous vector spaces g h v

spherical functions on spherical varieties let bf x bf h backslash bf g be a homogeneous spherical variety for a split reductive group bf g over the integers frak o of a p adic field k and k bf g frak o a hyperspecial maximal compact subgroup of g bf g k we compute eigenfunctions spherical functions on x bf x k under the action of the unramified or spherical hecke algebra of g generalizing many classical results of casselman shalika type under some additional assumptions on bf x we also prove a variant of the formula which involves a certain quotient of l values and we present several applications such as 1 a statement on good test vectors in the multiplicity free case namely that an h invariant functional on an irreducible unramified representation pi is non zero on pi k 2 the unramified plancherel formula for x including a formula for the tamagawa measure of bf x frak o and 3 a computation of the most continuous part of bf h period integrals of principal eisenstein series

automorphisms root systems and compactifications of homogeneous varieties let g be a complex semisimple group and let h c g be the group of fixed points of an involutive automorphism of g then x g h is called a symmetric variety in cp de concini and procesi have constructed an equivariant compactification x which has a number of remarkable properties some of them being i the boundary is the union of divisors d1 dr ii there are exactly 2r orbits their closures are the intersections di1 n ndi even schematically in particular there is only one closed orbit iii in case g is of adjoint type all orbit closures are smooth it is called the wonderful embedding of x or a complete symmetric variety and is the foundation for most deeper results about x independently luna and vust developed in lv a general theory of equivariant compactifications of homogeneous varieties under a connected reductive group g in particular they realized the reason which makes symmetric varieties behave so nicely a borel subgroup b has an open dense orbit in g h varieties with this property are called spherical luna and vust were able to describe all equivariant compactifications of them in terms of combinatorial data very similar to torus embeddings which are actually a special case they obtained in particular that every spherical embedding has only finitely many orbits nevertheless the reason for the existence of a compactification with properties i iii remained mysterious then brion and pauer established a relation with the automorphism group they proved in bp a spherical variety x g h possesses an equivariant compactification with exactly one closed orbit if and only if autg x ng h h is finite in this case there is a unique one which dominates all others the wonderful compactification x they also showed that the orbits of x correspond to the faces of a strictly convex polyhedral cone z then properties i and ii above are equivalent to z being simplicial this fact is much deeper and was proved by brion in brl in fact he showed much more let f be the set of characters of b which are the characters of a rational b eigenfunction on x this is a finitely generated free abelian group then the cone z is a subset of the real vector space hom f r brion showed that there is a finite reflection group wx acting on f such that z is one of its weyl chambers in case of a symmetric variety wx is its little weyl group

convergence rates for rank based models with applications to portfolio theory we determine rates of convergence of rank based interacting diffusions and semimartingale reflecting brownian motions to equilibrium bounds on fluctuations of additive functionals are obtained using transportation cost information inequalities for markov processes we work out various applications to the rank based abstract equity markets used in stochastic portfolio theory for example we produce quantitative bounds including constants for fluctuations of market weights and occupation times of various ranks for individual coordinates another important application is the comparison of performance between symmetric functionally generated portfolios and the market portfolio this produces estimates of probabilities of beating the market

first fundamental theorem for covariants of classical groups abstract let u g be a maximal unipotent subgroup of one of the classical groups g gl v o v sp v let w be a direct sum of copies of v and its dual v for the natural action u g w we describe a minimal system of homogeneous generators for the algebra of u g invariant regular functions on w for g o v sp v this result is connected with a construction for the irreducible representations of g due to h weyl

the extremal process of branching brownian motion we prove that the extremal process of branching brownian motion in the limit of large times converges weakly to a cluster point process the limiting process is a randomly shifted poisson cluster process where the positions of the clusters is a poisson process with intensity measure with exponential density the law of the individual clusters is characterized as branching brownian motions conditioned to perform unusually large displacements and its existence is proved the proof combines three main ingredients first the results of bramson on the convergence of solutions of the kolmogorov petrovsky piscounov equation with general initial conditions to standing waves second the integral representations of such waves as first obtained by lalley and sellke in the case of heaviside initial conditions third a proper identification of the tail of the extremal process with an auxiliary process based on the work of chauvin and rouault which fully captures the large time asymptotics of the extremal process the analysis through the auxiliary process is a rigorous formulation of the cavity method developed in the study of mean field spin glasses

hierarchical exchangeability of pure states in mean field spin glass models the main result in this paper is motivated by the mezard parisi ansatz which predicts a very special structure for the distribution of spins in diluted mean field spin glass models such as the random k sat model at positive temperature using the fact that one can safely assume the validity of the ghirlanda guerra identities in these models we prove hierarchical exchangeability of pure states for the asymptotic gibbs measures which allows us to apply a representation result for hierarchically exchangeable arrays recently proved in austin and panchenko in probab theory relat fields 2013 comparing this representation with the predictions of the mezard parisi ansatz one can see that the key property still missing is that the multi overlaps between pure states depend only on their overlaps

the extremal process of critical points of the pure p spin spherical spin glass model recently sharp results concerning the critical points of the hamiltonian of the p spin spherical spin glass model have been obtained by means of moments computations in particular these moments computations allow for the evaluation of the leading term of the ground state i e of the global minimum in this paper we study the extremal point process of critical points that is the point process associated to all critical values in the vicinity of the ground state we show that the latter converges in distribution to a poisson point process of exponential intensity in particular we identify the correct centering of the ground state and prove the convergence in distribution of the centered minimum to a minus gumbel variable these results are identical to what one obtains for a sequence of i i d variables correctly normalized namely we show that the model is in the universality class of rem

smooth varieties up to a 1 homotopy and algebraic h cobordisms abstract we start to study the problem of classifying smooth proper varieties over a field k from the standpoint of a 1 homotopy theory motivated by the topological theory of surgery we discuss the problem of classifying up to isomorphism all smooth proper varieties having a specified a 1 homotopy type arithmetic considerations involving the sheaf of a 1 connected components lead us to introduce several different notions of connectedness in a 1 homotopy theory we provide concrete links between these notions connectedness of points by chains of affine lines and various rationality properties of algebraic varieties e g rational connectedness we introduce the notion of an a 1 h cobordism an algebro geometric analog of the topological notion of h cobordism and use it as a tool to produce non trivial a 1 weak equivalences of smooth proper varieties also we give explicit computations of refined a 1 homotopy invariants such as the a 1 fundamental sheaf of groups for some a 1 connected varieties we observe that the a 1 fundamental sheaf of groups plays a central yet mysterious role in the structure of a 1 h cobordisms as a consequence of these observations we completely solve the classification problem for rational smooth proper surfaces over an algebraically closed field while there exist arbitrary dimensional moduli of such surfaces there are only countably many a 1 homotopy types each uniquely determined by the isomorphism class of its a 1 fundamental sheaf of groups

a class of linear differential dynamical systems with fuzzy matrices abstract this paper investigates the first order linear fuzzy differential dynamical systems with fuzzy matrices we use a complex number representation of the level sets of the fuzzy system and obtain the solution by employing such representation it is applicable to practical computations and has also some implications for the theory of fuzzy differential equations we then present some properties of the 2 dimensional dynamical systems and their phase portraits some examples are considered to show the richness of the theory and we can clearly see that new behaviors appear we finally present some conclusions and new directions for further research in the area of fuzzy dynamical systems

subeigenvectors and supereigenvectors of fuzzy matrices abstract a fuzzy algebra is a triple b where b is a nonempty bounded linearly ordered set and a b max a b a b min a b for a b b a vector x is said to be a eigenvector of a square matrix a if a x x for some b the aim of the paper is to solve subeigenproblem and supereigenproblem for some b that is to find a solution x of a x x and a x x x is called subeigenvector and supereigenvector respectively the problems are related to and motivated by the similar problems of tropical linear algebra in this paper the properties of subeigenvectors and supereigenvectors are described and the values associated with subeigenvectors and supereigenvectors are characterized as a consequence efficient algorithms for checking all equivalent conditions are introduced

the parisi ultrametricity conjecture in this paper we prove that the support of a random measure on the unit ball of a separable hilbert space that satises the ghirlanda guerra identities must be ultrametric with probability one this implies the parisi ultrametricity conjecture in mean eld spin glass models such as the sherrington kirkpatrick and mixed p spin models for which gibbs measures are known to satisfy the ghirlanda guerra identities in the thermodynamic limit

structure of finite rsb asymptotic gibbs measures in the diluted spin glass models we suggest a possible approach to proving the mezard parisi formula for the free energy in the diluted spin glass models such as diluted k spin or random k sat model at any positive temperature in the main contribution of the paper we show that a certain small modification of the hamiltonian in any of these models forces all finite rsb asymptotic gibbs measures in the sense of the overlaps to satisfy the mezard parisi ansatz for the distribution of spins unfortunately what is still missing is a description of the general full rsb asymptotic gibbs measures if one could show that the general case can be approximated by finite rsb case in the right sense then one could a posteriori remove the small modification of the hamiltonian to recover the mezard parisi formula for the original model

a new look at independence the concentration of measure phenomenon in product spaces is a farreaching abstract generalization of the classical exponential inequalities for sums of independent random variables we attempt to explain in the simplest possible terms the basic concepts underlying this phenomenon the basic method to prove concentration inequalities and the meaning of several of the most useful inequalities

an algorithm for addressing the real interval eigenvalue problem in this paper we present an algorithm for approximating the range of the real eigenvalues of interval matrices such matrices could be used to model real life problems where data sets suffer from bounded variations such as uncertainties e g tolerances on parameters measurement errors or to study problems for given states the algorithm that we propose is a subdivision algorithm that exploits sophisticated techniques from interval analysis the quality of the computed approximation and the running time of the algorithm depend on a given input accuracy we also present an efficient c implementation and illustrate its efficiency on various data sets in most of the cases we manage to compute efficiently the exact boundary points limited by floating point representation

uniqueness property for spherical homogeneous spaces let g be a connected reductive group recall that a homogeneous g space x is called spherical if a borel subgroup b g has an open orbit on x to x one assigns certain combinatorial invariants the weight lattice the valuation cone and the set of b stable prime divisors we prove that two spherical homogeneous spaces with the same combinatorial invariants are equivariantly isomorphic further we recover the group of g equivariant automorphisms of x from these invariants

single level constraint interval arithmetic in the present paper we propose a variant of constraint interval arithmetic that operates with a single parameter level in each interval operand of an expression this results in a computationally simple interval arithmetic that has several desirable properties not shared in general by interval operations previously defined in the literature the actualization and usefulness of this approach to fuzzy interval arithmetic is accomplished via alpha levels the evaluation of arithmetic expressions using the proposed single level constrained arithmetic is discussed in detail and we extend this theory to the fuzzy context

optimal transport maps for monge kantorovich problem on loop groups let g be a compact lie group and consider the loop group leg l c 0 1 g l 0 l 1 e let be the heat kernel measure at the time 1 for any density function f on leg such that ent f we shall prove that there exists a unique optimal transportation map t leg leg which pushes forward to f

monge kantorovitch measure transportation and monge ampere equation on wiener space let w h be an abstract wiener space assume two i i 1 2 probabilities on w w we give some conditions for the wasserstein distance between 1 and 2 with respect to the cameron martin space to be finite where the infimum is taken on the set of probability measures on w w whose first and second marginals are 1 and 2 in this case we prove the existence of a unique cyclically monotone map t iw with w h such that t maps 1 to 2 moreover if 2 then t is stochastically invertible i e there exists s w w such that s t iw 1 a s and t s iw 2 a s if in addition 1 then there exists a 1 convex function in the gaussian sobolev space such that these results imply that the quasi invariant transformations of the wiener space with finite wasserstein distance from can be written as the composition of a transport map t and a rotation i e a measure preserving map we give also 1 convex sub solutions and ito type solutions of the monge ampere equation on w

interval linear systems as a necessary step in fuzzy linear systems this article clarifies what it means to solve a system of fuzzy linear equations relying on the fact that they are a direct extension of interval linear systems of equations already studied in a specific interval mathematics literature we highlight four distinct definitions of a systems of linear equations where coefficients are replaced by intervals each of which based on a generalization of scalar equality to intervals each of the four extensions of interval linear systems has a corresponding solution set whose calculation can be carried out by a general unified method based on a relatively new concept of constraint intervals we also consider the smallest multidimensional intervals containing the solution sets we propose several extensions of the interval setting to systems of linear equations where coefficients are fuzzy intervals this unified setting clarifies many of the anomalous or inconsistent published results in various fuzzy interval linear systems studies

generalized hukuhara differentiability of interval valued functions and interval differential equations abstract in the present paper we use a recently proposed generalization of the hukuhara difference for compact convex sets to introduce and study a generalization of the hukuhara differentiability to the case of interval valued functions we also consider several possible definitions for the derivative of an interval valued function recently proposed in the literature and we study connections between them and their properties using these concepts we study interval differential equations local existence and uniqueness of two solutions is obtained together with characterizations of the solutions of an interval differential equation by ode systems several examples are also shown

equilibrium statistical mechanics of bipartite spin systems the aim of this paper is to give an extensive treatment of bipartite mean field spin systems pure and disordered at first bipartite ferromagnets are investigated and an explicit expression for the free energy is achieved through a new minimax variational principle then via the hamilton jacobi technique the same structure of the free energy is obtained together with the existence of its thermodynamic limit and the minimax principle is connected to a standard max one the same is investigated for bipartite spin glasses by the borel cantelli lemma we obtain the control of the high temperature regime while via the double stochastic stability technique we also obtain the explicit expression of the free energy in the replica symmetric approximation uniquely defined by a minimax variational principle again we also obtain a general result that states that the free energies of these systems are convex linear combinations of their independent one party model counterparts for the sake of completeness we show further that at zero temperature the replica symmetric entropy becomes negative and consequently such a symmetry must be broken the treatment of the fully broken replica symmetry case is deferred to a forthcoming paper as a first step in this direction we start deriving the linear and quadratic constraints to overlap fluctuations

weylgruppe und momentabbildung let g be a connected reductive group defined over an algebraically closed field of characteristic zero we assign to any g variety x a finite cristallographic reflection group w x by means of the moment map on the cotangent bundle this generalizes the little weyl group of a symmetric space the weyl group w x is related to the equivariant compactification theory of x we determine the closure of the image of the moment map and the generic isotropy group of the action of g on the cotangent bundle as a byproduct we determine the ideal of elements of mathfrak u g which act trivially on x as a differential operator

the interval eigenvalue problem using constraint interval analysis with an application to linear differential equations abstract we study the interval eigenvalue problem and its application to stability theory of interval linear differential equations the interval eigenvalue problem is studied from a constraint interval representation affording advantages not found in previous approaches