uniqueness of gradient gibbs measures with disorder we consider in a uniformly strictly convex potential regime two versions of random gradient models with disorder in model a the interface feels a bulk term of random fields while in model b the disorder enters through the potential acting on the gradients we assume a general distribution on the disorder with uniformly bounded finite second moments it is well known that for gradient models without disorder there are no gibbs measures in infinite volume in dimension d 2 while there are shift invariant gradient gibbs measures describing an infinite volume distribution for the gradients of the field as was shown by funaki and spohn commun math phys 185 1 36 1997 van enter and kulske proved in ann appl probab 18 1 109 119 2008 that adding a disorder term as in model a prohibits the existence of such gradient gibbs measures for general interaction potentials in d 2 in cotar and kulske ann appl probab 22 5 1650 1692 2012 we proved the existence of shift covariant random gradient gibbs measures for model a when d ge 3 the disorder is i i d and has mean zero and for model b when d ge 1 and the disorder has a stationary distribution in the present paper we prove existence and uniqueness of shift covariant random gradient gibbs measures with a given expected tilt u in mathbb r d and with the corresponding annealed measure being ergodic for model a when d ge 3 and the disordered random fields are i i d and symmetrically distributed and for model b when d ge 1 and for any stationary disorder dependence structure we also compute for both models for any gradient gibbs measure constructed as in cotar and kulske ann appl probab 22 5 1650 1692 2012 when the disorder is i i d and its distribution satisfies a poincare inequality assumption the optimal decay of covariances with respect to the averaged over the disorder gradient gibbs measure

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

delocalization of two dimensional random surfaces with hard core constraints we study the fluctuations of random surfaces on a two dimensional discrete torus the random surfaces we consider are defined via a nearest neighbor pair potential which we require to be twice continuously differentiable on a possibly infinite interval and infinity outside of this interval no convexity assumption is made and we include the case of the so called hammock potential when the random surface is uniformly chosen from the set of all surfaces satisfying a lipschitz constraint our main result is that these surfaces delocalize having fluctuations whose variance is at least of order log n where n is the side length of the torus we also show that the expected maximum of such surfaces is of order at least log n the main tool in our analysis is an adaptation to the lattice setting of an algorithm of richthammer who developed a variant of a mermin wagner type argument applicable to hard core constraints we rely also on the reflection positivity of the random surface model the result answers a question mentioned by brascamp et al on the hammock potential and a question of velenik

the proper formula for relative entropy and its asymptotics in quantum probability umegaki s relative entropys trd logd logd of states and with density operatorsd andd respectively is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint it is also proved that some other versions of the relative entropy give rise to the same asymptotics as umegaki s one as a byproduct the inequality tra logab tra loga logb is obtained for positive definite matricesa andb

quantum communication with coherent states quantum communication is studied under the cutoff rate criterion for transmission of m coherent states of real amplitude m 2 7 comparison with other measurement operators of interest such as the number and the quasi classical operators is made the effect of quantization of the decision level on the cutoff rate is considered

profile of blow up solutions to the exponential neumann boundary value problem abstract arising in problems of prescribed gaussian and geodesic curvatures we consider a blow up sequence of solutions to the following exponential neumann boundary value problem u n v n x x 2 n e 2 u n in b r u n h n x x n e u n on b r r 2 where denotes the outer unit normal vector to b r r 2 and n 1 we establish a uniform estimate for this blow up sequence of solutions near an isolated singular blow up point we treat two cases n and n separately and obtain two different uniform pointwise estimates

on compound state and mutual information in quantum information theory corresp a quantum mechanical compound state of an input state and its output state generated through a communication channel is constructed the mutual information of quantum communication theory is defined by using the compound state and its fundamental properties are studied

on the absence of breakdown of symmetry for the plane rotator model with long range random interaction we study the plane rotator model with hamiltonian frac 1 2 sum limits x ne y frac jxy cos mathcal o x mathcal o y left x y right 3 mathcal e where jxy for different pair x y are independent symmetric unbounded random variables it is proved that for almost all j all gibbs states p j are invariant by rotation

clustering in low dimensional so n invariant statistical models with long range interactions we obtain upper bounds for the two point correlation functions in statistical models in one or two dimensions which haveso n symmetry this clarifies upper bounds for long range interactions for which there exists clustering

energy entropy inequalities for classical lattice systems a new characterization of equilibrium states for classical lattice systems is given in terms of correlation inequalities their physical meaning is found to express thermodynamic stability we demonstrate the applicability of the inequalities in specific models

classification and construction of quantum communication systems we consider quantum communication systems qcs s introduced by davies 10 on trace class operators on general state spaces these are different from the usual quantum information theory as they deal with input and output being continuous in time we introduce the concept of a refinement of a qcs offering the possibility of distinguishing particles for instance according to their phase or location and classify refinements of general qcs we then introduce the notions of a bounded interaction rate and a bounded modulation and give in special cases a classification of qcs s satisfying both conditions as this classification shows the existence of a class of qcs s being completely different from the examples studied by davies we are able to discuss the cases of an independent detector and a linear modulation

relative entropy under mappings by stochastic matrices the relative g entropy of two finite discrete probability distributions x x1 xn and y y1 yn is defined as hg x y kxkg yk kk 1 where g 1 r is convex and g 0 0 when g t log 1 t then hg x y kxklog xk yk the usual relative entropy let pn x rn ixi 1 xi 0 i our major results is that for any m n column stochastic matrix a the contraction coefficient defined as g a sup hg ax ay hg x y x y pn x y satisfies g a 1 a where a minj k i min aij aik is dobrushin s coefficient of ergodicity consequently g a 1 if and only if a is scrambling upper and lower bounds on g a are established analogous results hold for markov chains in continuous time

strict convexity of the free energy for a class of non convex gradient models we consider a gradient interface model on the lattice with interaction potential which is a non convex perturbation of a convex potential we show using a one step multiple scale analysis the strict convexity of the surface tension at high temperature this is an extension of funaki and spohn s result 8 where the strict convexity of potential was crucial in their proof

continuity and relative hamiltonians let n n 1 be a norm convergent sequence of normal states on a von neumann algebraa with n let kn n 1 be a strongly convergent sequence of self adjoint elements ofa withkn k it is shown that the sequence omega n k n n geqq 1 of perturbed states converges in norm to a related result holds forc algebras a counter example is provided to show that it is not sufficient to assume weak convergence of n n 1 even whenkn k for alln however conditions are given which together with weak convergence are sufficient relative entropy methods are used and a relative entropy inequality is proved

on the relative entropy fora any subset of the bounded operators on a hilbert space containing the unit and and restrictions of states on toa ent a the entropy of relative to given the information ina is defined and given an axiomatic characterisation it is compared with ent the relative entropy introduced by umegaki and generalised by various authors which is defined only fora an algebra it is proved that ent and ent s agree on pairs of normal states on an injective von neumann algebra it is also proved that ent always has all the most important properties known for ent s monotonicity concavity w upper semicontinuity etc

on equivalence of spin and field pictures of lattice systems we investigate the spin and field systems on a lattice connected by the kac siegert transform it is shown that the structures of corresponding theories are equivalent in the sense of isomorphy of space of gibbs states and order parameters using the idea of equivalence of spin and field pictures we exhibit a class of lattice systems possessing infinitely uncountably many ground states the systems of this type with infinite range slow decaying interactions are expected to have a spin glass phase transition

spectral gap and decay of correlations in u 1 symmetric lattice systems in dimensions d 2 we consider many body systems with a global u 1 symmetry on a class of lattices with the fractal dimension d 2 and their zero temperature correlations whose observables behave as a vector under the u 1 rotation for a wide class of the models we prove that if there exists a spectral gap above the ground state then the correlation functions have a stretched exponentially decaying upper bound this is an extension of the mcbryan spencer method at finite temperatures to zero temperature the class includes quantum spin and electron models on the lattices and our method also allows finite or infinite quasi degeneracy of the ground state the resulting bounds rule out the possibility of the corresponding magnetic and electric long range orders

the liouville equation with singular data a concentration compactness principle via a local representation formula abstract for a bounded domain r 2 we establish a concentration compactness result for the following class of singular liouville equations u e u 4 j 1 m j p j in where pj j 0 and pj denotes the dirac measure with pole at point pj j 1 m our result extends brezis merle s theorem comm partial differential equations16 1991 1223 1253 concerning solution sequences for the regular liouville equation where the dirac measures are replaced by lp data p 1 in some particular case we also derive a mass quantization principle in the same spirit of li shafrir indiana univ math j 43 1994 1255 1270 our analysis was motivated by the study of the bogomol nyi equations arising in several self dual gauge field theories of interest in theoretical physics such as the chern simons theory self dual chern simons theories lecture notes in physics vol 36 springer verlag berlin 1995 and the electroweak theory selected papers on gauge theory of weak and electromagnetic interactions world scientific singapore

a generalized skew information and uncertainty relation a generalized skew information is defined and a generalized uncertainty relation is established with the help of a trace inequality which was recently proven by fujii in addition we prove the trace inequality conjectured by luo and zhang finally we point out that theorem 1 in s luo and q zhang ieee trans inf theory vol 50 pp 1778 1782 no 8 aug 2004 is incorrect in general by giving a simple counter example

asymptotic blow up analysis for singular liouville type equations with applications abstract we generalize the pointwise estimates obtained in 2 19 and 34 concerning blow up solutions of the liouville type equation u x 2 w x e u in with r 2 open and bounded 1 and w any lipschitz continuous function which satisfies 0 a w b we focus to the case left open in 2 and 34 where the parameter n whose analysis is much more involved as we need to resolve the difficulty of a genuinely non radial behaviour of blow up solutions in the worst situation there is no chance in general to resolve the profile in the form of a solution of a liouville equation in r 2 instead we need to adopt iterated blow up arguments next we refine our blow up analysis to cover a class of planar liouville type problems see 1 27 1 28 below arising from the study of cosmic strings cfr 28 35 in this context we are able to distinguish between a single blow up radial profile and the case of multiple blow up profiles typical of non radial solutions as a consequence we obtain a radial symmetry result which is interesting in itself but also contributes towards the sharp solvability issue for the planar problem 1 27 1 28

on the absence of spontaneous symmetry breaking and of crystalline ordering in two dimensional systems we develop a unified approach based on araki s relative entropy concept to proving absence of spontaneous breaking of continuous internal symmetries and translation invariance in two dimensional statistical mechanical systems more precisely we show that under rather general assumptions on the interactions all equilibrium states of a two dimensional system have all the symmetries compact internal and spatial of the dynamics except possibly rotation invariance rotation invariance remains unbroken if connected correlations decay more rapidly than the inverse square distance we also prove that two dimensional systems with a non compact internal symmetry group like anharmonic crystals typically do not have gibbs states

the golden thompson trace inequality is complemented we prove a class of trace inequalities which complements the golden thompson inequality for example tr epa epb 2 p tr ea b holds for all p 0 when a and b are hermitian matrices and denotes the geometric mean we also prove related trace inequalities involving the logarithmic function namely p 1tr x log yp 2xpyp 2 tr x log x log y p 1tr x log xp 2ypxp 2 for all p 0 when x and y are nonnegative matrices these inequalities supply lower and upper bounds on the relative entropy

inertia groups and fibers let x and y be proper normal connected schemes over a field k and let f x y be a finite flat k morphism which is generically galois i e the extension of function fields k y k x is galois with galois group g it is well known that for the zariski open complement u y of the branch locus of f the map f 1 u u is a right g torsor thus for any y u and x f 1 y the extension of fields k x k y is galois and the stabilizer in g of x maps isomorphically to the galois group gal k x k y in particular when the fiber f 1 y is irreducible then gal k x k y g if k is a global field it is natural to ask how the injection gal k x k y g relates arithmetic inertia groups in gal k x k y with geometric inertia groups in g corresponding to ramification in the map f the same question can be asked more generally when k is the function field of a connected normal noetherian scheme s with positive dimension where arithmetic ramification in gal k x k y corresponds to ramification in k x of the valuations on k y arising from codimension 1 points of the normalization of s in k y a special case of this question was investigated by s beckmann she considered the case when k is a number field with integer ring ok y pk and x is a geometrically connected curve over k let a1 am be the finitely many branch points of f since k has characteristic 0 so f is tamely ramified over each ai the inertia groups of f over the ai s are cyclic subgroups of g for any closed point y pk it is not difficult to show that the scheme theoretic closure y in p1ok which is proper over ok is also quasi finite and therefore finite over ok for example if y y pk are distinct closed points then y y is artinian in particular when y is a k rational point distinct from the ai s the intersection y ai is an artinian closed subscheme of y spec ok let ip y ai 0 denote the length of the part of y ai which lies over p spec ok so obviously ip y ai 0 for all but finitely many p depending on y and ai let f denote the finite set of primes p of ok at which one of the following occurs some k ai k is ramified at p ip ai aj 0 for some i 6 j i e the closures ai and aj in p1ok meet over p the p t adic valuation on k pk k t is ramified in k x p divides the degree of f note that f can be effectively determined and depends only on the the geometry of f and the arithmetic in some of the fibers of f beckmann proved the following result

a new proof of ruelle s superstability bounds for a slightly stronger assumptions on the interaction we give a very transparent proof of ruelle s result in the language of poisson integral measure representation for the correlation functions on the configuration space using some kind of cluster expansion in the densities of configurations

u 1 su 2 gauge invariance of non relativistic quantum mechanics and generalized hall effects we show that the non relativistic quantum mechanics of particles with spin coupled to an electromagnetic field has a naturalu 1 su 2 gauge invariance ward identities reflecting this gauge invariance combined with an assumption of incompressibility of a system of such particles in an appropriate external field and for suitable values of the particle density permit us to determine the form of the effective action of the system as a functional of small fluctuations in the electromagnetic field in the large distance adiabatic limit in this limit the action is found to have a universal form we present explicit results for two dimensional incompressible electron fluids and apply them to derive the equations of linear response theory describing a variety of generalized hall effects sum rules for the hall conductivities magnetic susceptibilities and other quantities of physical interest are found

counting number fields in fibers with an appendix by jean gillibert let x be a projective curve over mathbb q and t in mathbb q x a non constant rational function of degree n ge 2 for every tau in mathbb z pick p tau in x bar mathbb q such that t p tau tau dvornicich and zannier proved that for large n the field mathbb q p 1 ldots p n is of degree at least e cn log n over mathbb q where c 0 depends only on x and t in this paper we extend this result replacing mathbb q by an arbitrary number field

inequalities for quantum divergences and the audenaert datta conjecture given two density matrices rho and sigma there are a number of different expressions that reduce to the alpha renyi relative entropy of rho with respect to sigma in the classical case i e when rho and sigma commute only those expressions for which the data processing inequality dpi is valid are of potential interest as quantum divergences in quantum information theory audenaert and datta have made a conjecture on the validity of the dpi for an interesting family of quantum generalizations of the alpha renyi relative entropies the alpha z renyi relative entropies they and others have contributed to the partial solution of this conjecture we review the problem its context and the methods that have been used to obtain the results that are known at present presenting a unified treatment of developments that have unfolded in a number of different papers

on the symmetry of the gibbs states in two dimensional lattice systems under fairly general conditions if a two dimensional classical lattice system has an internal symmetry groupg which is a compact connected lie group then all gibbs states areg invariant

multivariate trace inequalities we prove several trace inequalities that extend the golden thompson and the araki lieb thirring inequality to arbitrarily many matrices in particular we strengthen lieb s triple matrix inequality as an example application of our four matrix extension of the golden thompson inequality we prove remainder terms for the monotonicity of the quantum relative entropy and strong sub additivity of the von neumann entropy in terms of recoverability we find the first explicit remainder terms that are tight in the commutative case our proofs rely on complex interpolation theory as well as asymptotic spectral pinching providing a transparent approach to treat generic multivariate trace inequalities

euclidean invariance in statistical mechanics of classical continuous system it is shown that equilibrium states of classical particles with short range interactions are euclidean invariant whenever their correlation functions have a clustering which is integrable the relation between invariance and clustering is analysed for spatial rotations and internal rotational degrees of freedom the analysis is then extended to the case of long range interactions including the coulomb force and jellium systems

large deviations and variational principle for harmonic crystals we consider massless gaussian fields with covariance related to the green function of a long range random walk on d these are viewed as gibbs measures for a linear quadratic interaction we establish thermodynamic identities and prove a version of gibbs variational principle showing that translation invariant gibbs measures are characterized as minimizers of the relative entropy density we then study the large deviations of the empirical field of a gibbs measure we show that a weak large deviation principle holds at the volume order with rate given by the relative entropy density

the mermin wagner phenomenon and cluster properties of one and two dimensional systems we give optimal conditions concerning the range of interactions for the absence of spontaneous breakdown of continuous symmetries for one and two dimensional quantum and classical lattice and continuum systems for a class of models verifying infrared bounds our conditions are necessary and sufficient using the same techniques we obtain a priori bounds on clustering for systems with continuous symmetry improving results of jasnow and fisher

some properties of correlations of quantum lattice systems in thermal equilibrium simple proofs of uniqueness of the thermodynamic limit of kms states and of the decay of equilibrium correlations are presented for a large class of quantum lattice systems at high temperatures new quantum correlation inequalities for general heisenberg models are described finally a simplified derivation of a general result on power law decay of correlations in 2d quantum lattice systems with continuous symmetries is given extending results of mcbryan and spencer for the 2d classical xy model

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

quantum ergodic channels in operator algebras abstract the quantum ergodic channel is studied by operator algebraic methods the ergodic and kms channels are introduced and their dynamical properties are discussed

operational capacity and pseudoclassicality of a quantum channel we explore some basic properties of coding theory of a general quantum communication channel and its operational capacity including 1 adaptive measurement with feedback code 2 reconsideration of single letterized capacity formula and 3 pseudoclassicality of a channel

absence of phase transitions in certain one dimensional long range random systems an ising chain is considered with a potential of the formj i j i j where thej i j are independent random variables with mean zero the chain contains both randomness and frustration and serves to model a spin glass a simple argument is provided to show that the system does not exhibit a phase transition at a positive temperature if 1 this is to be contrasted with a ferromagnetic interaction which requires 2 the basic idea is to prove that the surface free energy between two half lines is finite although the surface energy may be unbounded ford dimensional systems it is shown that the free energy does not depend on the specific boundary conditions if 1 2 d

operator monotones the reduction criterion and the relative entropy we introduce the theory of operator monotone functions and employ it to derive a new inequality relating the quantum relative entropy and the quantum conditional entropy we present applications of this new inequality and in particular we prove a new lower bound on the relative entropy of entanglement and other properties of entanglement measures

adversarial hypothesis testing and a quantum stein s lemma for restricted measurements recall the classical hypothesis testing setting with two sets of probability distributions p and q one receives either n i i d samples from a distribution p in p or from a distribution q in q and wants to decide from which set the points were sampled it is known that the optimal exponential rate at which errors decrease can be achieved by a simple maximum likelihood ratio test which does not depend on p or q but only on the sets p and q we consider an adaptive generalization of this model where the choice of p in p and q in q can change in each sample in some way that depends arbitrarily on the previous samples in other words in the k th round an adversary having observed all the previous samples in rounds 1 ldots k 1 chooses p k in p and q k in q with the goal of confusing the hypothesis test we prove that even in this case the optimal exponential error rate can be achieved by a simple maximum likelihood test that depends only on p and q we then show that the adversarial model has applications in hypothesis testing for quantum states using restricted measurements for example it can be used to study the problem of distinguishing entangled states from the set of all separable states using only measurements that can be implemented with local operations and classical communication locc the basic idea is that in our setup the deleterious effects of entanglement can be simulated by an adaptive classical adversary we prove a quantum stein s lemma in this setting in many circumstances the optimal hypothesis testing rate is equal to an appropriate notion of quantum relative entropy between two states in particular our arguments yield an alternate proof of li and winter s recent strengthening of strong subadditivity for von neumann entropy

blow up analysis for solutions to neumann boundary value problem abstract in this paper we will analyze the blow up behavior of solution sequences satisfying a conformal invariant equation defined on a compact 2 dimensional surface m g with boundary we will provide some accurate point wise estimates for the profile of these sequences

quantum energy entropy inequalities a new method for proving the absence of symmetry breaking for quantum systems we develop a new method based on a general energy entropy inequality to rule out spontaneous breaking of symmetries the main advantage of our scheme consists in its clear cut physical significance and its new areas of applicability in particular we can handle discrete symmetry groups as well as continuous ones finally a few illustrations are discussed

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

a mermin wagner theorem for gibbs states on lorentzian triangulations we establish a mermin wagner type theorem for gibbs states on infinite random lorentzian triangulations lt arising in models of quantum gravity such a triangulation is naturally related to the distribution p of a critical galton watson tree conditional upon non extinction at the vertices of the triangles we place classical spins taking values in a torus m of dimension d with a given group action of a torus g of dimension d d in the main body of the paper we assume that the spins interact via a two body nearest neighbor potential u x y invariant under the action of g we analyze quenched gibbs measures generated by u and prove that for p almost all lorentzian triangulations every such gibbs measure is g invariant which means the absence of spontaneous continuous symmetry breaking

on estimating the derivatives of symmetric diffusions in stationary random environment with applications to ϕ interface model we consider diffusions on d or random walks on d in a random environment which is stationary in space and in time and with symmetric and uniformly elliptic coefficients we show existence and holder continuity of second space derivatives and time derivatives for the annealed kernels of such diffusions and give estimates for these derivatives in the case of random walks these estimates are applied to the ginzburg landau interface model

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

relative hamiltonians which are not bounded from above abstract let be a normal state on a von neumann algebra a and let h be a self adjoint operator affiliated with a which is bounded below we discuss the perturbed state h h is defined when it exists to be the unique state maximizing the function ent a h where ent a is the entropy of relative to we prove that given a state with x for some x r then there exists a generalized operator h with h we prove that h h h h whenever the right hand side exists we prove that if is faithful and h h for which h is a kms state with respect to the modular group of is a dense subset of the set of states kms with respect to the modular group of h

the global markov property for equilibrium states which are determined by correlations in a strip consider an interaction round face potential on the lattice d which may include nearest neighbor and next nearest neighbor pair interactions as well as more general plaquette terms assuming some periodicity of the potential it is shown that under the condition that equilibrium states can be distinguished by expectation values of sufficiently local observables equilibrium states possess a global markov property the condition under which this markov property is shown to hold is met in particular in case the equilibrium state is unique or determined by the magnetization the proof is based on an application of the variational principle to states which are constructed like markov chains