linear response theory for thermally driven quantum open systems this note is a continuation of our recent paper v jaksic y ogata and c a pillet the green kubo formula and onsager reciprocity relations in quantum statistical mechanics commun math phys in press where we have proven the green kubo formula and the onsager reciprocity relations for heat fluxes in thermally driven quantum open systems in this note we extend the derivation of the green kubo formula to heat and charge fluxes and discuss some other generalizations of the model and results of v jaksic y ogata and c a pillet the green kubo formula and onsager reciprocity relations in quantum statistical mechanics commun math phys in press

backward perturbation analysis of certain characteristic subspaces this paper gives optimal backward perturbation bounds and the accuracy of approximate solutions for subspaces associated with certain eigenvalue problems such as the eigenvalue problemax x the generalized eigenvalue problem ax bx and the singular value decomposition of a matrixa this paper also gives residual bounds for certain eigenvalues generalized eigenvalues and singular values

a preparation theorem for weierstrass systems it is shown that lion and rolin s preparation theorem for globally subanalytic functions holds for the collection of definable functions in any expansion of the real ordered field by a weierstrass system

on two functions of a matrix with positive definite hermitian part let a be a square complex matrix with positive definite hermitian part h a a a h 2 and let f a g a be two functions defined by f a h a 1 1 g a a 1 a h this paper derives

on eigenvalue variations of rayleigh quotient matrix pencils of a definite pencil abstract let a b be a definite matrix pencil of order n i e both a and b are n n hermitian and c a b def min xe c n x 2 1 x h a ib x 0 suppose y is an n l matrix with full column rank whose column vectors span an approximate invariant subspace for a b this note investigates the relation between eigenvalues of a b and those of y h ay y h by our result for the spectral norm improves sun s linear algebra appl 139 253 267 1990 we also present bounds in frobenius norms and in general unitarily invariant norms

constructible exponential functions motivic fourier transform and transfer principle we introduce spaces of exponential constructible functions in the motivic setting for which we construct direct image functors in the absolute and relative settings this allows us to define a motivic fourier transformation for which we get various inversion statements we also define spaces of motivic schwartz bruhat functions on which motivic fourier transformation induces isomorphisms our motivic integrals specialize to nonarchimedean integrals we give a general transfer principle comparing identities between functions defined by exponential integrals over local fields of characteristic zero resp of positive characteristic having the same residue field we also prove new results about p adic integrals of exponential functions and stability of this class of functions under p adic integration

valuation theory of exponential hardy fields i we describe the valuation theoretic properties of the hardy fields associated to models of t exp where t is the theory of a polynomially bounded o minimal expansion of the reals and exp is the real exponential function we deduce that t exp has levels with parameters and is exponentially bounded we establish a maximality property of h mathbb r rm an powers the hardy field of the expansion by the restricted analytic functions and power functions

a limited feedback joint precoding for amplify and forward relaying this paper deals with the practical precoding design for a dual hop downlink with multiple input multiple output mimo amplify and forward relaying first assuming that full channel state information csi of the two hop channels is available a suboptimal dual hop joint precoding scheme i e precoding at both the base station and relay station is investigated based on its structure a scheme of limited feedback joint precoding using joint codebooks is then proposed which uses a distributed codeword selection to concurrently choose two joint precoders such that the feedback delay is considerably decreased finally the joint codebook design for the limited feedback joint precoding system is analyzed and results reveal that independent codebook designs at the base station and relay station using the conventional grassmannian subspace packing method is able to guarantee that the overall performance of the dual hop joint precoding scheme improves with the size of each of the two codebooks simulation results show that the proposed dual hop joint precoding system using distributed codeword selection scheme exhibits a rate or ber performance close to the one using the optimal centralized codeword selection scheme while having lower computational complexity and shorter feedback delay

wielandt and ky fan theorem for matrix pairs the generalization of wielandt and ky fan theorem is given for hermitian matrix pairs and some new eigenvalue perturbation estimates are obtained an application is made on a class of quadratic matrix pencils

definable sets in ordered structures i this paper introduces and begins the study of a well behaved class of linearly ordered structures the minimal structures the definition of this class and the corresponding class of theories the strongly dlminlmal theories is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories theorems 2 1 and 2 3 respectively provide characterizations of 41 minimal ordered groups and rings several other simple results are collected in 3 the primary tool in the analysis of minimal structures is a strong analogue of ii forking symmetry given by theorem 4 2 this result states that any parametri cally definable unary function in an yminimal structure is piecewise either constant or an ore er preserving or reversing bijection of intervals the results that follow include the existence and uniqueness of prime models over sets theorem 5 1 and a characterization of all bto categorical aminimal structures theorem 6 1 1 introduction the class of linearly ordered structures has long been an im portant subject of concern to model theorists impressive results have been obtained in the study of models of several particular theories that extend the theory of linear order among those theories that have been approached successfully are peano arithmetic the theory of ordered abelian groups that of real closed fields and that of linear order itself yet very little has been done in the way of developing a general model theory for ordered structures in this paper we develop the model theory for a class of linearly ordered structures that we isolate by demanding that a structure in this class satisfy a condition whose effect is that the linear ordering and the algebraic part of the structure behave quite well with respect to one another let l be a finitary first order language and w an l structure a set of n tuples a c x is said to be parametrically definable if there is some l formula

symmetric matrix pencils abstract a significant number of matrix eigenvalue problems quadratic or linear are best reformulated as pencils a m in which both a and m are real and symmetric some examples are given and then the canonical forms are re examined to explain the role of the sign characteristic attached to real eigenvalues in addition we examine the limitations on the use of the rayleigh quotient functional x ax x mx in describing the eigenvalues this sheds new light on the class of definite pencils and the stability of their eigenvalues under perturbations the reduction of indefinite pencils to useful sparse forms is mentioned

music like doa estimation without estimating the number of sources this paper presents a new direction of arrival doa estimation algorithm where doa estimation is realized in the framework of beamforming by introducing a new optimization problem the new optimization problem aims to minimize the variance of the array output subject to retaining the output power at the look direction and precluding a trivial solution the proposed algorithm is equivalent to the multiple signal classification music algorithm in some special cases it is therefore named as the music like method however different from the classical music algorithm the new method is not based on subspace decomposition and the number of sources is hence not required for direction finding this enables us to avoid performance deterioration caused by incorrect source number estimation the effectiveness and advantage of the proposed algorithm is theoretically investigated and computational complexity is also addressed to verify the theoretical analysis computer simulations are implemented and comparisons with other algorithms are made

master symmetries of the xy model master symmetries found by barouch and fuchssteiner for a finite size xy model with the help of a computer program are mathematically analyzed for an infinitely extended xy model by a rigorous operator algebraic method with an easy computation the infinite family of commuting hamiltonians and the master symmetries generating them form an infinite dimensional lie group of automorphisms of ac algebra of observables for the model

a note on stewart s theorem for definite matrix pairs abstract let a and b be n n hermitian matrices the matrix pair a b is called definite pair and the corresponding eigenvalue problem ax bx is definite if c a b inf x 1 h a ib x 0 in this note we develop a uniform upper bound for differences of corresponding eigenvalues of two definite pairs and so improve a result which is obtained by g w stewart 2 moreover we prove that this upper bound is a projective metric in the set of n n definite pairs

complements of subanalytic sets and existential formulas for analytic functions we show that the complement of a subanalytic set defined by real analytic functions from any subalgebra closed under differentiation is a subanalytic set defined by the functions from the same subalgebra this result has an equivalent formulation in logic consider an expression built from functions as above using equalities and inequalities as well as existential and universal quantifiers such an expression is equivalent to an existential expression involving functions from the same class provided that the variables approach neither infinity nor the boundary of the domain

ilyashenko algebras based on transserial asymptotic expansions abstract we construct a hardy field that contains ilyashenko s class of germs at of almost regular functions found in 12 as well as all log exp analytic germs this implies non oscillatory behaviour of almost regular germs with respect to all log exp analytic germs in addition each germ in this hardy field is uniquely characterized by an asymptotic expansion that is an le series as defined by van den dries et al 7 as these series generally have support of order type larger than the notion of asymptotic expansion itself needs to be generalized

on perturbations of matrix pencils with real spectra a revisit this paper continues earlier studies by bhatia and li on eigenvalue perturbation theory for diagonalizable matrix pencils having real spectra a unifying framework for creating crucial perturbation equations is developed with the help of a recent result on generalized commutators involving unitary matrices new and much sharper bounds are obtained

the stability of the non equilibrium steady states we show that the non equilibrium steady state ness of the free lattice fermion model far from equilibrium is macroscopically unstable the problem is translated to that of the spectral analysis of the liouville operator we use the method of positive commutators to investigate it we construct a positive commutator on the lattice fermion system whose dispersion relation is k cosk

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

stability behavior for unsupervised learning abstract we consider models proposed by oja karhunen cox and adams for a task of unsupervised learning questions of existence of solutions and stability behavior of those solutions are addressed we also discuss the impact of temporary connections in the stability of the learning process

broken translation invariance in quasifree fermionic correlations out of equilibrium abstract using the c algebraic scattering approach to study quasifree fermionic systems out of equilibrium in quantum statistical mechanics we construct the nonequilibrium steady state in the isotropic xy chain whose translation invariance has been broken by a local magnetization and analyze the asymptotic behavior of the expectation value for a class of spatial correlation observables in this state the effect of the breaking of translation invariance is twofold mathematically the finite rank perturbation not only regularizes the scalar symbol of the invertible toeplitz operator generating the leading order exponential decay but also gives rise to an additional trace class hankel operator in the correlation determinant physically in its decay rate the nonequilibrium steady state exhibits a left mover right mover structure affected by the scattering at the impurity

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

ax lindemann for mathcal a _g we prove the ax lindemann theorem for the coarse moduli space a g of principally polarized abelian varieties of dimension g 1 we affirm the andre oort conjecture unconditionally for a g for g 6 and under grh for all g

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

generalized eigenvalues of a definite hermitian matrix pair abstract in this note we study some basic properties of generalized eigenvalues of a definite hermitian matrix pair in particular we prove an interlacing theorem and a minimax theorem we also obtain upper bounds for the variation of the generalized eigenvalues under perturbation these results extend and improve those of r c li j g sun and g w stewart on the topic

relative eigenvalues of a definite matrix pair abstract let a and b be hermitian matrices and let c a b min x 2 1 xh a ib x the matrix pair a b is called a definite pair and the corresponding eigenvalue problem ax bx is definite if c a b 0 the relationship between the eigenvalues of a b and those of a definite pair of the form xh1ax1 xh1bx1 is studied in this paper the eigenvalues of xh1ax1 xh1bx1 are said to be the relative eigenvalues of a b with respect to the subspace r x1 from the main result of this paper one can deduce a corresponding conclusion on hermitian matrices

estimation of the eigenvalues of ab for a 0 b 0 abstract for positive definite hermitian matrices a and b we obtain an upper and a lower bound for the eigenvalues of ab in terms of those of a and b

stability under integration of sums of products of real globally subanalytic functions and their logarithms we study lebesgue integration of sums of products of globally subanalytic functions and their logarithms called constructible functions our first theorem states that the class of constructible functions is stable under integration the second theorem treats integrability conditions in fubini type settings and the third result gives decay rates at infinity for constructible functions further we give preparation results for constructible functions related to integrability conditions

non equilibrium steady states of the xy chain we study the non equilibrium statistical mechanics of the two sided xy chain we start from an initial state in which the left x m part of the lattice are at inverse temperatures betal and betar using a simple scattering theoretic analysis we construct the unique non equilibrium steady state ness this state depends on betal and betar but not on the choice of the decoupling parameter m we prove that in the non equilibrium case betal betar this state has strictly positive entropy production

on perturbations of matrix pencils with real spectra perturbation bounds for the generalized eigenvalue problem of a diagonalizable matrix pencil a ab with real spectrum are developed it is shown how the chordal distances between the generalized eigenvalues and the angular distances between the generalized eigenspaces can be bounded in terms of the angular distances between the matrices the applications of these bounds to the spectral variations of definite pencils are conducted in such a way that extra attention is paid to their peculiarities so as to derive more sophisticated perturbation bounds our results for generalized eigenvalues are counterparts of some celebrated theorems for the spectral variations of hermitian matrices such as the weyl lidskii theorem and the hoffman wielandt theorem and those for generalized eigenspaces are counterparts of the celebrated davis kahan sin 0 sin 20 theorems for the eigenspace variations of hermitian matrices the paper consists of two parts part i is for generalized eigenvalue perturbations while part ii deals with generalized eigenspace perturbations

perturbation theorems for the generalized eigenvalue problem abstract given a matrix pair z a b the perturbation of its eigenvalues is studied considering two pairs z w as points of the grassman manifold g n 2 n and its eigenvalues as points in g 1 2 the projective complex plane the distance of the spectra measured in the chordal metric in g 1 2 is bounded by some distance of the matrix pairs in g n 2 n analogs of the bauer fike theorem henrici s theorem and the hoffman weilandt theorem are obtained from which the classical results can be derived

lebesgue classes and preparation of real constructible functions abstract we call a function constructible if it has a globally subanalytic domain and can be expressed as a sum of products of globally subanalytic functions and logarithms of positively valued globally subanalytic functions for any q 0 and constructible functions f and on e r n we prove a theorem describing the structure of the set x p e 0 f x l p x q where x q is the positive measure on r n whose radon nikodym derivative with respect to the lebesgue measure is x q y x y q we also prove a closely related preparation theorem for f and these results relate analysis the study of l p spaces with geometry the study of zero loci

the elementary theory of restricted analytic fields with exponentiation numbers with exponentiation is model complete when we combine this with hovanskii s finiteness theorem 9 it follows that the real exponential field is o minimal in o minimal expansions of the real field the definable subsets of r share many of the nice structural properties of semialgebraic sets for example definable subsets have only finitely many connected components definable sets can be stratified and triangulated and continuous definable maps are piecewise trivial see 5 in this paper we will prove a quantifier elimination result for the real field augmented by exponentiation and all restricted analytic functions and use this result to obtain o minimality we were led to this while studying work of ressayre 13 and several of his ideas emerge here in simplified form however our treatment is formally independent of the results of 16 17 9 and 13

norms of certain matrices with applications to variations of the spectra of matrices and matrix pencils abstract we start by proving a lower bound for the lp operator norm of a submatrix with sufficiently large dimensions and some relations between the lp norm and the lp operator norm we then study the lp norm and the lp operator norm of certain matrices arising in estimating the eigenvalue generalized eigenvalue singular value and generalized singular value variations of matrices matrix pencils and matrix pairs finally as applications of our bounds upon the lp norm and lp operator norm of certain matrices we give several new bounds for the variations of the spectra of matrices and matrix pencils these new bounds are generalizations of the weyl lidskii theorem for hermitian matrices and the bhatia davis theorem for unitary matrices and of some results obtained by the author recently for diagonalizable pencils with real spectra

the perturbation bounds for eigenspaces of a definite matrix pair leta andb be hermitian matrices the matrix pair a b is called definite pair and the corresponding eigenvalue problem ax bx is definite if c a b equiv mathop min limits left x right 1 x h a ib x 0 the perturbation bounds for eigenspaces of a definite pair on every unitary invariant matrix norm were obtained by imposing additional restrictions on the location of the generalized eigenvalues thus it gives a positive answer for an open question proposed by stewart 7 the famous davis kahan sin theorems and sin 2 theorem 2 can also be deduced from the present results

the green kubo formula and the onsager reciprocity relations in quantum statistical mechanics we study linear response theory in the general framework of algebraic quantum statistical mechanics and prove the green kubo formula and the onsager reciprocity relations for heat fluxes generated by temperature differentials our derivation is axiomatic and the key assumptions concern ergodic properties of non equilibrium steady states

randomizing a model abstract a randomization of a first order structure m is a new structure with certain closure properties whose universe is a setkof random elements of m randomizations assign probabilities to sentences of the language of m with new constants fromk our main theorem shows that all randomizations of m are models of the same first order theoryt which has a nice set of axioms and admits elimination of quantifiers moreover the class of substructures of models oftis characterized by a natural setvof universal axioms oft so thattis the model completion ofv

trace minimization principles for positive semi definite pencils abstract this paper is concerned with x h ax subject to x h bx j for a hermitian matrix pencil a b where j is diagonal and j 2 i the identity matrix of apt size the same problem was investigated earlier by kovac striko and veselic linear algebra appl 216 1995 139 158 for the case in which b is assumed nonsingular but in this paper b is no longer assumed nonsingular and in fact a b is even allowed to be a singular pencil it is proved among others that the infimum is finite if and only if a b is a positive semi definite pencil in the sense that there is a real number 0 such that a 0 b is positive semi definite the infimum when finite can be expressed in terms of the finite eigenvalues of a b sufficient and necessary conditions for the attainability of the infimum are also obtained

integration of semialgebraic functions and integrated nash functions when integrating semialgebraic functions one has to leave the semialgebraic setting for example one gets the global logarithm and iterated antiderivatives of algebraic power series such as the arctangent we show that it is enough to enlarge the semialgebraic functions by these functions to completely describe parameterized integrals of semialgebraic functions to realize this we close the rings of algebraic power series in arbitrary dimension under taking antiderivatives we analyze these rings profoundly in particular we show that the weierstrass division theorem and the weierstrass preparation theorem hold this allows us to apply model theoretic results to obtain an explicit description of parameterized integrals of semialgebraic functions finally we investigate the structure generated by the integrated algebraic power series

on perturbations of matrix pencils with real spectra ii a well known result on spectral variation of a hermitian matrix due to mirsky is the following let a and a be two n x n hermitian matrices and let 1 n and 1 n be their eigenvalues arranged in ascending order then diag 1 1 n n a a for any unitarily invariant norm in this paper we generalize this to the perturbation theory for diagonalizable matrix pencils with real spectra the much studied case of definite pencils is included in this

inclusion dynamics hybrid automata hybrid systems are dynamical systems with the ability to describe mixed discrete continuous evolution of a wide range of systems consequently at first glance hybrid systems appear powerful but recalcitrant neither yielding to analysis and reasoning through a purely continuous time modeling as with systems of differential equations nor open to inferential processes commonly used for discrete state transition systems such as finite state automata a convenient and popular model called hybrid automata was introduced to model them and has spurred much interest on its tractability as a tool for inference and model checking in a general setting intuitively a hybrid automaton is simply a finite state automaton with each state augmented by continuous variables which evolve according to a set of well defined continuous laws each specified separately for each state this article investigates both the notion of hybrid automaton and the model checking problem over such a structure in particular it relates first order theories and analysis results on multivalued maps and reduces the bounded reachability problem for hybrid automata whose continuous laws are expressed by inclusions x f x t to a decidability problem for first order formulaeover the reals furthermore the paper introduces a class of hybrid automata for which the reachability problem can be decided and shows that the problem of deciding whether a hybrid automaton belongs to this class can be again decided using first order formulaeover the reals despite the fact that the bisimulation quotient for this class of hybrid automata can be infinite we show that our techniques permit effective model checking for a nontrivial fragment of ctl

quasianalytic denjoy carleman classes and o minimality we show that the expansion of the real field generated by the functions of a quasianalytic denjoy carleman class is model complete and o minimal provided that the class satisfies certain closure conditions some of these structures do not admit analytic cell decomposition and they show that there is no largest o minimal expansion of the real field