improved estimation of the disturbance variance in a linear regression model abstract in this paper we consider estimation of the disturbance variance in a linear regression model a new class of improved estimators is obtained by extending results dating to stein 1964 these estimators dominate the ordinary least squares estimator under squared error loss

semidirect products and reduction in mechanics this paper shows how to reduce a hamiltonian system on the cotangent bundle of a lie group to a hamiltonian system in the dual of the lie algebra of a semidirect product the procedure simplifies unifies and extends work of greene guillemin holm holmes kupershmidt marsden morrison ratiu sternberg and others the heavy top compressible fluids magnetohydrodynamics elasticity the maxwell vlasov equations and multifluid plasmas are presented as examples starting with lagrangian variables our method explains in a direct way why semidirect products occur so frequently in examples it also provides a framework for the systematic introduction of clebsch or canonical variables

integrable xyz spin chain with boundaries we consider a general class of boundary terms of the open xyz spin 1 2 chain compatible with integrability we have obtained the general elliptic solution of a k matrix obeying the boundary yang baxter equation using the r matrix of the eight vertex model and derived the associated integrable spin chain hamiltonian

relationship between minimum gap and success probability in adiabatic quantum computing we explore the relationship between two figures of merit for an adiabatic quantum computation process the success probability p and the minimum gap min between the ground and first excited states investigating to what extent the success probability for an ensemble of problem hamiltonians can be fitted by a function of min and the computation time t we study a generic adiabatic algorithm and show that a rich structure exists in the distribution of p and min in the case of two qubits p is to a good approximation a function of min of the stage in the evolution at which the minimum occurs and of t this structure persists in examples of larger systems

bayesian prediction tests for structural stability abstract a bayesian prediction test for structural stability is proposed which has several advantages over classical procedures in particular the small sample distribution of the test statistic is known under the null hypothesis the same test statistic is appropriate for both regression and time series models the test is valid even for nonstationary multivariate ar and var models and the test is robust to certain forms of nonnormality in the error distribution a comparison of the bayesian and classical tests indicates that the bayesian test is equivalent to the predictive chow test for regression models

implementation of group covariant positive operator valued measures by orthogonal measurements we consider group covariant positive operator valued measures povms on a finite dimensional quantum system following neumark s theorem a povm can be implemented by an orthogonal measurement on a larger system accordingly our goal is to find a quantum circuit implementation of a given group covariant povm which uses the symmetry of the povm based on representation theory of the symmetry group we develop a general approach for the implementation of group covariant povms which consist of rank one operators the construction relies on a method to decompose matrices that intertwine two representations of a finite group we give several examples for which the resulting quantum circuits are efficient in particular we obtain efficient quantum circuits for a class of povms generated by weyl heisenberg groups these circuits allow to implement an approximative simultaneous measurement of the position and crystal momentum of a particle moving on a cyclic chain

the complexity of solving equations over finite groups we study the computational complexity of solving systems of equations over a finite group an equation over a group g is an expression of the form w1 w2 wk 1g where each wi is either a variable an inverted variable or a group constant and 1g is the identity element of g a solution to such an equation is an assignment of the variables to values in g which realizes the equality a system of equations is a collection of such equations a solution is then an assigmnent which simultaneously realizes each equation we show that the problem of determining if a single equation has a solution is np complete for all nonsolvable groups g for nilpotent groups this same problem is shown to be in p the analogous problem for systems of such equations is shown to be np complete if g is non abelian and in p otherwise finally we observe some connections between these problems and the theory of nonuniform automata

efficient computation of the fourier transform on finite groups let g be a finite group f g c a function and p an ir reducible representation of g the fourier transform is defined as f p eseeg f s p s direct computation for all irreducible representations involves order igi operations we derive fast algorithms and develop them for the symmetric group sn there n 2 is reduced to nf nf al2 where a is the constant for matrix multiplication 2 38 as of this writing variations of the algorithm allow efficient computation for small representations a practical version of the algorithm is given on sn numerical evidence is presented to show a speedup by a factor of 100 for n 9 department of mathematics harvard university cambridge massachusetts 02138 department of mathematics columbia university new york new york 10027 this content downloaded from 157 55 39 153 on mon 19 sep 2016 04 53 00 utc all use subject to http about jstor org terms

adiabatic quantum computation is equivalent to standard quantum computation the model of adiabatic quantum computation is a relatively recent model of quantum computation that has attracted attention in the physics and computer science communities we describe an efficient adiabatic simulation of any given quantum circuit this implies that the adiabatic computation model and the standard circuit based quantum computation model are polynomially equivalent our result can be extended to the physically realistic setting of particles arranged on a two dimensional grid with nearest neighbor interactions the equivalence between the models allows one to state the main open problems in quantum computation using well studied mathematical objects such as eigenvectors and spectral gaps of hamiltonians

generalized tops and moment maps to loop algebras using a combination of hamiltonian reduction under continuous and discrete symmetry groups the lax equations for the motion of generalized tops are derived as part of a general approach to moment map embeddings of integrable hamiltonian systems in loop algebras

pre testing for linear restrictions in a regression model with spherically symmetric disturbances abstract in this paper we derive the exact risk under quadratic loss of pre test estimators of the prediction vector and of the error variance of a linear regression model with spherically symmetric disturbances the pre test in question is one of the validity of a set of exact linear restrictions on the model s coefficient vector we demonstrate how the known results for the model with normal disturbances can be extended to this broader case we also show that the critical value of unity results in a minimum of the risk of the pre test estimator of the error variance to illustrate the results we assume multivariate student t regression disturbances and numerically evaluate the derived expressions

estimation of the error variance after a preliminary test of homogeneity in a regression model with spherically symmetric disturbances abstract in this paper we consider the risk under quadratic loss of an estimator of the error variance after a pre test for homogeneity of the variances in the two sample linear regression model we investigate the effects on risk of assuming normal disturbances when in fact the error distribution is spherically symmetric we also broaden the standard assumption that the never pool variance estimators are based on the least squares principle to illustrate the results numerically we consider two families of variance mixing distributions one of which results in multivariate student t disturbances our results show that in some situations we should always pre test even if the error variances are equal and we provide the optimal test critical value the evaluations also show that using the least squares technique to form the never pool estimators may not always be the preferred strategy

quantum twisted yangians symmetry baxterisation and centralizers based on the quantum twisted yangians integrable systems with special boundary conditions called soliton nonpreserving snp may be constructed in the present article we focus on the study of subalgebras of the quantum twisted yangians and we show that such a subalgebra provides an exact symmetry of the rational transfer matrix we discuss how the spectrum of a generic transfer matrix may be obtained by focusing only on two types of special boundaries it is also shown that the subalgebras emerging from the asymptotics of tensor product representations of the quantum twisted yangian turn out to be dual to the quantum brauer algebra to deal with general boundaries in the trigonometric case we propose a new algebra which also provides the appropriate framework for the baxterisation procedure in the snp case

bayesian marginal equivalence of elliptical regression models the use of proper prior densities in regression models with multivariate non normal elliptical error distributions is examined when the scale matrix is known up to a precision factor t treated as a nuisance parameter marginally equivalent models preserve the convenient predictive and posterior results on the parameter of interest b obtained in the reference case of the normal model and its conditionally natural conjugate gamma prior prior densities inducing this property are derived for two special cases of non normal elliptical densities representing very different patterns of tail behavior in a linear framework so called semi conjugate prior structures are defined as leading to marginal equivalence to a normal data density with a fully natural conjugate prior

integrable impurities for an open fermion chain employing the graded versions of the yang baxter equation and the reflection equations we construct two kinds of integrable impurities for a small polaron model with general open boundary conditions a we shift the spectral parameter of the local lax operator at arbitrary sites in the bulk and b we embed the impurity fermion vertex at each boundary of the chain the hamiltonians with different types of impurity terms are given explicitly the bethe ansatz equations as well as the eigenvalues of the hamiltonians are constructed by means of the quantum inverse scattering method in addition we discuss the ground state properties in the thermodynamic limit

on the kowalevski goryachev chaplygin gyrostat the invariant separated variables are constructed for the kowalevski goryachev chaplygin gyrostat using decomposition of the corresponding lagrangian submanifold on a symmetric product of two copies of single spectral curve of the 2 2 lax matrix a generalization to the bundle of the poisson brackets is discussed

representations of the quadratic algebra and partially asymmetric diffusion with open boundaries we consider the one dimensional partially asymmetric exclusion model with open boundaries the model describes a system of hard core particles that hop stochastically in both directions with different rates at both boundaries particles are injected and extracted by means of the method of derrida et al the stationary probability measure can be expressed as a matrix product state involving two matrices forming a fock like representation of a general quadratic algebra we obtain the representations of this algebra which were unknown in the mathematical literature and use the two dimensional one to derive exact expressions for the density profile and correlation functions using the correspondence between the stochastic model and a quantum spin chain we obtain exact correlation functions for a spin heisenberg xxz chain with non diagonal boundary terms generalizations to other reaction diffusion models are discussed

optimal critical values of pre tests when estimating the regression error variance analytical findings under a general loss structure this paper re visits the problem of estimating the regression error variance in a linear multiple regression model after preliminary hypothesis tests for either linear restrictions on the coefficients or homogeneity of variances there is an extensive literature that discusses these problems particularly in terms of the sampling properties of the pre test estimators using various loss functions as the basis for risk analysis in this paper a unified framework for analysing the risk properties of these estimators is developed under a general class of loss structures that incorporates virtually all first order differentiable losses particular consideration is given to the choice of critical values for the pre tests analytical results indicate that an level substantially higher than those normally used may be appropriate for optimal risk properties under a wide range of loss functions the paper also generalizes some known analytical results in the pre test literature and proves other results only previously shown numerically

fusion procedure for the belavin model with open boundary conditions we have recently used unitarity and cross unitarity properties of an symmetric r matrix to construct the transfer matrix t u for an n site open spin chain here we give a fusion procedure for such a chain and we prove that the fused transfer matrix is commutative with the original transfer matrix t u

integrable systems on so 4 related to xxx spin chains with boundaries we consider two site xxx heisenberg magnets with different boundary conditions which are integrable systems on so 4 possessing additional cubic and quartic integrals of motion the separated variables for these models are constructed using the sklyanin method

boundary conditions for integrable quantum systems a new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering r matrix method the method proposed allows the author to treat open quantum chains with appropriate boundary terms in the hamiltonian the general considerations are applied to the xxz and xyz models the nonlinear schrodinger equation and toda chain

robust bayesian inference in elliptical regression models broadening the stochastic assumptions on the error terms of regression models was prompted by the analysis of linear multivariate t models in zellner 1976 we consider a possibly nonlinear regression model under any multivariate elliptical data density and examine bayesian posterior and predictive results the latter are shown to be robust with respect to the specific choice of a sampling density within this elliptical class in particular sufficient conditions for such model robustness are that we single out a precision factor 2 on which we can specify an improper prior density apart from theb posterior distribution of this nuisance parameter 2 the entire analysis will then be completely unaffected by departures from normality similar results hold in finite mixtures of such elliptical densities which can be used to average out specification uncertainty

upper bound on the characters of the symmetric groups let c be a conjugacy class in the symmetric group s n and be a partition of n let f be the degree of the irreducible representation s c the character of s at c and r c the normalized character c f we prove that there exist constants b 0 and 1 q 0 such that for n 4 for every conjugacy class c in s n and every irreducible representation s of s n r c max q 1 n 1 n b supp c where supp c is the number of non fixed digits under the action of a permutation in c 1 is the size of the largest part in and 1 is the number of parts in the proof is obtained by enumeration of rim hook tableaux the hook formula and probabilistic arguments combinatorial algebraic and statistical applications follow this result in particular we estimate the rate of mixing of random walks on the alternating groups with respect to conjugacy classes

outlier models and prior distributions in bayesian linear regression summary bayesian inference in regression models is considered using heavy tailed error distri butions to accommodate outliers the particular class of distributions that can be con structed as scale mixtures of normal distributions are examined and use is made of them as both error models and prior distributions in bayesian linear modelling includ ing simple regression and more complex hierarchical models with structured priors depending on unknown hyperprior parameters the modelling of outliers in nominally normal linear regression models using alternative error distributions which are heavy tailed relative to the normal provides an automatic means of both detecting and accommodating possibly aberrant observations such realistic models do however often lead to analytically intractable analyses with complex posterior distributions in several dimensions that are difficult to summarize and understand in this paper we consider a special yet rather wide class of heavy tailed unimodal and symmetric error distributions for which the analyses though apparently intractable can be examined in some depth by exploiting certain properties of the assumed error form the distributions concerned are those that can be con structed as scale mixtures of normal distributions in his paper concerning location parameters de finetti 1961 discusses such distributions and suggests the hypothetical interpretation that each observation is taken using an instrument with normal error but each time chosen at random from a collection of instruments of different precisions the distribution of the

integrability of two interacting n dimensional rigid bodies a new class of integrable euler equations on the lie algebra so 2n describing twon dimensional interacting rigid bodies is found a lax representation of equations of motion which depends on a spectral parameter is given and complete integrability is proved the double hamiltonian structure and the lax representation of the general flow is discussed

nonstandard comodules for quantum matrix bialgebras we carry out a construction of nonstandard matrix quantum groups comodules through a generalization of the coaction on tensor powers of a comodule this generalization amounts to consider the situation where quantum space coordinates and its endomorphism matrix entries belong to a noncommutative quadratic algebra

a special case of neumann s system and the kowalewski chaplygin goryachev top the authors present l operators both for the special case of neumann s system and the kowalewski chaplygin goryachev top these are quantum systems integrable by the quantum inverse scattering r matrix method l operators being 2 2 matrices satisfy an algebra generated by an r matrix of the xxx type the close connection between the two models is demonstrated they carry out a non obvious separation of variables and also give a dynamical group scheme for the eigenstate problem of neumann s system this separation differs from those in euler angles and allows them to find eigenenergies in an effective way

on the coalescence time of reversible random walks consider a system of coalescing random walks where each individual performs random walk over a finite graph g or more generally evolves according to some reversible markov chain generator q let c be the first time at which all walkers have coalesced into a single cluster c is closely related to the consensus time of the voter model for this g or q we prove that the expected value of c is at most a constant multiple of the largest hitting time of an element in the state space this solves a problem posed by aldous and fill and gives sharp bounds in many examples including all vertex transitive graphs we also obtain results on the expected time until only k 2 clusters remain our proof tools include a new exponential inequality for the meeting time of a reversible markov chain and a deterministic trajectory which we believe to be of independent interest

hidden symmetry subgroup problems we advocate a new approach for addressing hidden structure problems and finding efficient quantum algorithms we introduce and investigate the hidden symmetry subgroup problem hssp which is a generalization of the well studied hidden subgroup problem hsp given a group acting on a set and an oracle whose level sets define a partition of the set the task is to recover the subgroup of symmetries of this partition inside the group the hssp provides a unifying framework that besides the hsp encompasses a wide range of algebraic oracle problems including quadratic hidden polynomial problems while the hssp can have provably exponential quantum query complexity we obtain efficient quantum algorithms for various interesting cases to achieve this we present a general method for reducing the hssp to the hsp which works efficiently in several cases related to symmetries of polynomials the hssp therefore connects in a rather surprising way certain hidden polynomial problems with the hsp using this co

ground state spaces of frustration free hamiltonians we study the ground state space properties for frustration free hamiltonians we introduce a concept of reduced spaces to characterize local structures of ground state spaces for a many body system we characterize mathematical structures for the set k of all the k particle reduced spaces which with a binary operation called join forms a semilattice that can be interpreted as an abstract convex structure the smallest nonzero elements in k called atoms are analogs of extreme points we study the properties of atoms in k and discuss its relationship with ground states of k local frustration free hamiltonians for spin 1 2 systems we show that all the atoms in 2 are unique ground states of some 2 local frustration free hamiltonians moreover we show that the elements in k may not be the join of atoms indicating a richer structure for k beyond the convex structure our study of k deepens the understanding of ground state space properties for frustration free hamiltonians from the new perspective of reduced spaces

bayes prediction in regressions with elliptical errors abstract in this paper the prediction problem is considered for linear regression models with elliptical errors when the bayes prior is non informative we show that the bayes prediction density under the elliptical errors assumption is exactly the same as that obtained with normally distributed errors thus assuming that the errors have a normal distribution when the true distribution is elliptical will not lead to incorrect predictive inferences if the error variance structure is correctly specified this extends the results of zellner 1976 finally based on monte carlo numerical integration procedures computations are provided in a model with multiplicative heteroscedasticity

the complexity of the local hamiltonian problem the k locham problem is a natural complete problem for the complexity class qma the quantum analogue of np it is similar in spirit to sc max k sat which is np complete for k geq 2 it was known that the problem is qma complete for any k geq 3 on the other hand 1 locham is in p and hence not believed to be qma complete the complexity of the 2 locham problem has long been outstanding here we settle the question and show that it is qma complete we provide two independent proofs our first proof uses only elementary linear algebra our second proof uses a powerful technique for analyzing the sum of two hamiltonians this technique is based on perturbation theory and we believe that it might prove useful elsewhere using our techniques we also show that adiabatic computation with 2 local interactions on qubits is equivalent to standard quantum computation

a quantum walk inspired adiabatic algorithm for solving graph isomorphism problems we present a quantum algorithm for solving graph isomorphism problems that is based on an adiabatic protocol we use a collection of continuous time quantum walks each one generated by an xy hamiltonian to visit the configuration space in this way we avoid a diffusion over all the possible configurations and significantly reduce the dimensionality of the accessible hilbert space within this restricted space the graph isomorphism problem can be translated into searching for a satisfying assignment to a 2 sat satisfiable formula and mapped onto a 2 local hamiltonian without resorting to perturbation gadgets or projective techniques we present an analysis of the time for execution of the algorithm on small graph isomorphism problem instances and discuss the issue of an implementation of the proposed adiabatic scheme on current quantum computing hardware

approximate predictive likelihood summary a predictive likelihood is given which approximates both bayes and maximum likeli hood predictive inference by expansion of a posterior likelihood this synthesizes and extends previous results and is widely applicable the approximation usually differs from exact bayes posterior predictive density by op n 2 and from exact predictive likelihood by op n 1 but does not depend on the availability of prior information and is applicable when exact predictive likelihood cannot be found the results are applied to the prediction various treatments of the likelihood function form the basis of parametric inference but until recently few non bayesian attempts have been made to define versions of it suitable for prediction however lauritzen 1974 hinkley 1979 and more recently butler 1986 have proposed rather similar definitions of predictive likelihood based on the idea of conditioning on the value of a sufficient statistic unfortunately they apply only when sufficiency provides a genuine reduction of the data moreover even if for a given problem an exact predictive likelihood exists the calculations needed to derive it can be onerous this paper provides approximations to hinkley lauritzen predictive likelihood accur ate to op n 1 in many cases which are applicable in situations where maximum likelihood estimation is regular they also apply to problems which admit no exact predictive likelihood because sufficiency provides no real reduction of the data the approximations are derived from a posterior predictive density and may if prior informa tion is available be regarded as bayesian procedures their construction requires only the repeated maximization of a likelihood and evaluation of the observed information matrix at the maximum and is fast and accurate using usual numerical maximization methods there is a close connexion with the approximations to posterior moments and marginal densities given by tierney kadane 1986 who suggest the same approxima tion as used here but give no detailed results for prediction section 2 of this paper shows how laplace s method for integrals may be applied to a posterior predictive density to yield approximate predictive likelihood section 3 discusses a connection with conditional inference essentially the same approximation

testing linear restrictions on coefficients in a linear regression model with proxy variables and spherically symmetric disturbances abstract in this paper we consider the power function of the classical f test for linear restrictions on the coefficients in a linear regression model with spherically symmetric disturbances when proxies are used in the place of unobservable regressors to illustrate the results we numerically evaluate the power function for two families of variance mixing distributions in a simple univariate regression model our results show the effects on the power function of the degree of nonnormality and of the correlation between the omitted and the proxy variable

complexity of stoquastic frustration free hamiltonians we study several problems related to properties of nonnegative matrices that arise at the boundary between quantum and classical probabilistic computation our results are twofold first we identify a large class of quantum hamiltonians describing systems of qubits for which the adiabatic evolution can be efficiently simulated on a classical probabilistic computer these are stoquastic local hamiltonians with a frustration free ground state a hamiltonian belongs to this class iff it can be represented as h sum a h a where 1 every term h a acts nontrivially on a constant number of qubits 2 every term h a has real nonpositive off diagonal matrix elements in the standard basis and 3 the ground state of h is a ground state of every term h a second we generalize the cook levin theorem proving np completeness of the satisfiability problem to the complexity class ma merlin arthur games a probabilistic analogue of np specifically we construct a quantum version of the k sat problem which we call stoquastic k sat such that stoquastic k sat is contained in ma for any constant k and any promise problem in ma is karp reducible to stoquastic 6 sat this result provides the first nontrivial example of a ma complete promise problem

estimating the error variance in regression after a preliminary test of restrictions on the coefficients abstract maximum likelihood estimation of the variance of the disturbances in a linear regression is considered in the context of exact linear restrictions on the coefficient vector the exact risk function for the associated preliminary test estimator of this variance is derived and evaluated numerically on the basis of relative quadratic loss our results indicate that in practical situations the risk of the pre test estimator differs only slightly from that of the naive estimator which ignores the restrictions without testing applying the coefficient restrictions without testing their validity is not recommended

a subexponential time quantum algorithm for the dihedral hidden subgroup problem we present a quantum algorithm for the dihedral hidden subgroup problem dhsp with time and query complexity 2 o sqrt log n in this problem an oracle computes a function f on the dihedral group d n which is invariant under a hidden reflection in d n by contrast the classical query complexity of dhsp is o sqrt n the algorithm also applies to the hidden shift problem for an arbitrary finitely generated abelian group the algorithm begins as usual with a quantum character transform which in the case of d n is essentially the abelian quantum fourier transform this yields the name of a group representation of d n which is not by itself useful and a state in the representation which is a valuable but indecipherable qubit the algorithm proceeds by repeatedly pairing two unfavorable qubits to make a new qubit in a more favorable representation of d n once the algorithm obtains certain target representations direct measurements reveal the hidden subgroup

new integrable problem of classical mechanics complete integrability in liouville s sense is proven for rotation of an arbitrary rigid body with a fixed point in a newtonian field with an arbitrary homogeneous quadratic potential a consequences is the complete integrability of rotation of a rigid body with fixed center of mass in the field of arbitrary sufficiently remote objects in the second approximation explicit formulae are obtained expressing angular velocities of the rigid body in terms of functions for riemannian surfaces integrable cases are found for rotation of a rigid body in nonlinear newtonian potential fields

constant solutions of reflection equations and quantum groups to the yang baxter equation an additional relation can be added this is the reflection equation that appears in various places with or without spectral parameter e g in factorizable scattering on a half line integrable lattice models with nonperiodic boundary conditions noncommutative differential geometry on quantum groups etc two forms of spectral parameter independent reflection equations are studied chosen by the requirement that their solutions be comodules with respect to the quantum group coaction leaving invariant the reflection equations for a variety of known solutions of the yang baxter equation the constant solutions of the reflection equations are given various quadratic algebras defined by the reflection equations are also given explicitly

the continuous multivariate location scale model revisited a tale of robustness summary we consider the location scale family of all multivariate continuous sampling distributions any model from the subclass induced by fixing a particular distribution on the unit sphere combined with a standard improper prior on the scale parameter will lead to the same inference on location and observables this finding sheds light on many robustness results in less general models and its implications are illustrated in a number of examples

ramsey numbers and adiabatic quantum computing the graph theoretic ramsey numbers are notoriously difficult to calculate in fact for the two color ramsey numbers r m n with m n 3 only nine are currently known we present a quantum algorithm for the computation of the ramsey numbers r m n we show how the computation of r m n can be mapped to a combinatorial optimization problem whose solution can be found using adiabatic quantum evolution we numerically simulate this adiabatic quantum algorithm and show that it correctly determines the ramsey numbers r 3 3 and r 2 s for 5 s 7 we then discuss the algorithm s experimental implementation and close by showing that ramsey number computation belongs to the quantum complexity class quantum merlin arthur

on the robustness of lm lr and w tests in regression models in this paper we show that in the small sample case while the lr test is robust the lm and w tests are not robust when the errors are student t for the large sample however all three tests are found to be robust a variation of the w test is also considered and it turns out to be nonrobust the implications of assuming the errors to be distributed as multivariate student t rather than multivariate normal are also discussed it is found that not all of evans and savin s 4 results carry through when the errors are student t

tensor network contractions for sat the computational cost of counting the number of solutions satisfying a boolean formula which is a problem instance of sat has proven subtle to quantify even when finding individual satisfying solutions is computationally easy e g 2 sat which is in mathsf p determining the number of solutions can be mathsf p hard recently computational methods simulating quantum systems experienced advancements due to the development of tensor network algorithms and associated quantum physics inspired techniques by these methods we give an algorithm using an axiomatic tensor contraction language for n variable sat instances with complexity o g cd o 1 2 c where c is the number of copy tensors g is the number of gates and d is the maximal degree of any copy tensor thus n variable counting problems can be solved efficiently when their tensor network expression has at most o log n copy tensors and polynomial fan out this framework also admits an intuitive proof of a variant of the tovey conjecture the r 1 sat instance of the dubois tovey theorem this study increases the theory expressiveness and application of tensor based algorithmic tools and provides an alternative insight on these problems which have a long history in statistical physics and computer science

the structure of simultaneous equation estimators a generalization towards nonnormal disturbances a general linear simultaneous equation system with a multivariate student t disturbance vector is considered the normal equations of the corresponding maximum likelihood estimator are used as estimator generating equations to introduce a new class of estimators properties of large subclasses of these estimators are determined for disturbance vectors other than the multivariate student t

testing conditional independence via rosenblatt transforms this paper proposes new tests of conditional independence of two random variables given a single index involving an unknown finite dimensional parameter the tests employ rosenblatt transforms and are shown to be distribution free while retaining computational convenience some results from monte carlo simulations are presented and discussed

a new integrable case of the motion of the 4 dimensional rigid body a lax pair for a new family of integrable systems on so 4 is presented the construction makes use of a twisted loop algebra of theg 2 lie algebra we also describe a general scheme producing integrable cases of the generalized rigid body motion in an external field which have a lax representation with spectral parameter several other examples of multi dimensional tops are discussed

quantum computational complexity of the n representability problem qma complete we study the computational complexity of the n representability problem in quantum chemistry we show that this problem is quantum merlin arthur complete which is the quantum generalization of nondeterministic polynomial time complete our proof uses a simple mapping from spin systems to fermionic systems as well as a convex optimization technique that reduces the problem of finding ground states to n representability

a note on bayesian inference in a regression model with elliptical errors in this note estimation and prediction is considered for a linear or nonlinear regression model with the error vector distributed as a scale mixture of multivariate normal distributions the results obtained for the linear model by zellner 1976 jammalamadaka et al 1987 and chib et al 1988 are explained and generalized to much more general classes of regression models and prior distributions

some matrix variate distribution theory notational considerations and a bayesian application summary we introduce and justify a convenient notation for certain matrix variate distributions which by its emphasis on the important underlying parameters and the theory on which it is based eases greatly the task of manipulating such distributions important examples include the matrix variate normal t f and beta and the wishart and inverse wishart distributions the theory is applied to compound matrix distributions and to bayesian prediction in the multivariate linear model

limitations of quantum coset states for graph isomorphism it has been known for some time that graph isomorphism reduces to the hidden subgroup problem hsp what is more most exponential speedups in quantum computation are obtained by solving instances of the hsp a common feature of the resulting algorithms is the use of quantum coset states which encode the hidden subgroup an open question has been how hard it is to use these states to solve graph isomorphism it was recently shown by moore et al 2005 that only an exponentially small amount of information is available from one or a pair of coset states a potential source of power to exploit are entangled quantum measurements that act jointly on many states at once we show that entangled quantum measurements on at least n log n coset states are necessary to get useful information for the case of graph isomorphism matching an information theoretic upper bound this may be viewed as a negative result because in general it seems hard to implement a given highly entangled measurement our main theorem is very general and also rules out using joint measurements on few coset states for some other groups such as gl n fpm and gn where g is finite and satisfies a suitable property

the optimal size of a preliminary test of linear restrictions in a misspecified regression model abstract when the choice of estimator for the coefficients in a linear regression model is determined by the outcome of a prior test of the validity of restrictions on the model it is well known that a minimax risk regret criterion leads to the simple rule that the optimal critical value for the preliminary test is approximately two in value regardless of the degrees of freedom we show that this result no longer holds in the likely event that relevant regressors are excluded from the model at the outset

a fast algorithm for approximating the ground state energy on a quantum computer estimating the ground state energy of a multiparticle system with relative error using deterministic classical algorithms has cost that grows exponentially with the number of particles the problem depends on a number of state variablesd that is proportional to the number of particles and suers from the curse of dimensionality quantum computers can vanquish this curse in particular we study a ground state eigenvalue problem and exhibit a quantum algorithm that achieves relative error using a number of qubits c 0 d log 1 with total cost number of queries plus other quantum operations cd 3 where 0 is arbitrarily small and c and c 0 are inde

geodesic flow on so 4 and the intersection of quadrics this note studies the relationship between the complete integrability of geodesic flow on so 4 with a left invariant metric and the geometry of the intersection of four quadrics in p 6

conditional and unconditional statistical independence abstract conditional independence almost everywhere in the space of the conditioning variates does not imply unconditional independence although it may well imply unconditional independence of certain functions of the variables an example that is important in linear regression theory is discussed in detail this involves orthogonal projections on random linear manifolds which are conditionally independent but not unconditionally independent under normality necessary and sufficient conditions are obtained under which conditional independence does imply unconditional independence

the hidden subgroup problem and quantum computation using group representations the hidden subgroup problem is the foundation of many quantum algorithms an efficient solution is known for the problem over abelian groups employed by both simon s algorithm and shor s factoring and discrete log algorithms the nonabelian case however remains open an efficient solution would give rise to an efficient quantum algorithm for graph isomorphism we fully analyze a natural generalization of the algorithm for the abelian case to the nonabelian case and show that the algorithm determines the normal core of a hidden subgroup in particular normal subgroups can be determined we show however that this immediate generalization of the abelian algorithm does not efficiently solve graph isomorphism

new construction for a qma complete three local hamiltonian we present a new way of encoding a quantum computation into a three local hamiltonian our construction is novel in that it does not include any terms that induce legal illegal clock transitions therefore the weights of the terms in the hamiltonian do not scale with the size of the problem as in previous constructions this improves the construction by kempe and regev quantum inf comput 3 258 264 2003 e print quant ph 0302079 who were the first to prove that three local hamiltonian is complete for the complexity class qma the quantum analog of np quantum k sat a restricted version of the local hamiltonian problem using only projector terms was introduced by bravyi e print quant ph 0602108 as an analog of the classical k sat problem bravyi proved that quantum 4 sat is complete for the class qma with one sided error qma1 and that quantum 2 sat is in p we give an encoding of a quantum circuit into a quantum 4 sat hamiltonian using only three local terms as an intermediate step to this th

accurate approximations for posterior moments and marginal densities abstract this article describes approximations to the posterior means and variances of positive functions of a real or vector valued parameter and to the marginal posterior densities of arbitrary i e not necessarily positive parameters these approximations can also be used to compute approximate predictive densities to apply the proposed method one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima nevertheless the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third order expansions of the likelihood and requiring the evaluation of third derivatives the approximate marginal posterior densities behave very much like saddle point approximations for sampling distributions the principal regularity condition required is that the likelihood times prior be unimodal

bethe subalgebras in affine birman murakami wenzl algebras and flat connections for q kz equations commutative sets of jucys murphy elements for affine braid groups of types were defined construction of r matrix representations of the affine braid group of type and its distinguished commutative subgroup generated by the type jucys murphy elements are given we describe a general method to produce flat connections for the two boundary quantum knizhnik zamolodchikov equations as necessary conditions for sklyanin s type transfer matrix associated with the two boundary multicomponent zamolodchikov algebra to be invariant under the action of the type jucys murphy elements we specify our general construction to the case of the birman murakami wenzl algebras bmw algebras for short as an application we suggest a baxterization of the dunkl cherednik elements in the double affine hecke algebra of type a

vector nonlinear schrodinger equation on the half line we investigate the manakov model or more generally the vector nonlinear schrodinger equation on the half line using a backlund transformation method two classes of integrable boundary conditions are derived mixed neumann dirichlet and robin boundary conditions integrability is shown by constructing a generating function for the conserved quantities we apply a nonlinear mirror image technique to construct the inverse scattering method with these boundary conditions the important feature in the reconstruction formula for the fields is the symmetry property of the scattering data emerging from the presence of the boundary particular attention is paid to the discrete spectrum an interesting phenomenon of transmission between the components of a vector soliton interacting with the boundary is demonstrated this is specific to the vector nature of the model and is absent in the scalar case for one soliton solutions we show that the boundary can be used to make certain components of the incoming soliton vanishingly small this is reminiscent of the phenomenon of light polarization by reflection

testing for stochastic monotonicity we propose a test of the hypothesis of stochastic monotonicity this hypothesis is of interest in many applications in economics our test is based on the supremum of a rescaled u statistic we show that its asymptotic distribution is gumbel the proof is difficult because the approximating gaussian stochastic process contains both a stationary and a nonstationary part and so we have to extend existing results that only apply to either one or the other case we also propose a refinement to the asymptotic approximation that we show works much better in finite samples we apply our test to the study of intergenerational income mobility

product state approximations to quantum states we show that for any many body quantum state there exists an unentangled quantum state such that most of the two body reduced density matrices are close to those of the original state this is a statement about the monogamy of entanglement which cannot be shared without limit in the same way as classical correlation our main application is to hamiltonians that are sums of two body terms for such hamiltonians we show that there exist product states with energy that is close to the ground state energy whenever the interaction graph of the hamiltonian has high degree this proves the validity of mean field theory and gives an explicitly bounded approximation error if we allow states that are entangled within small clusters of systems but product across clusters then good approximations exist when the hamiltonian satisfies one or more of the following properties 1 high degree 2 small expansion or 3 a ground state where the blocks in the partition have sublinear entanglement previously this was known only in the case of small expansion or in the regime where the entanglement was close to zero our approximations allow an extensive error in energy which is the scale considered by the quantum pcp probabilistically checkable proof and nlts no low energy trivial state conjectures thus our results put restrictions on the possible hamiltonians that could be used for a possible proof of the qpcp or nlts conjectures by contrast the classical pcp constructions are often based on constraint graphs with high degree likewise we show that the parallel repetition that is possible with classical constraint satisfaction problems cannot also be possible for quantum hamiltonians unless qpcp is false the main technical tool behind our results is a collection of new classical and quantum de finetti theorems which do not make any symmetry assumptions on the underlying states

solving systems of diagonal polynomial equations over finite fields we present an algorithm to solve a system of diagonal polynomial equations over finite fields when the number of variables is greater than some fixed polynomial of the number of equations whose degree depends only on the degree of the polynomial equations our algorithm works in time polynomial in the number of equations and the logarithm of the size of the field whenever the degree of the polynomial equations is constant as a consequence we design polynomial time quantum algorithms for two algebraic hidden structure problems for the hidden subgroup problem in certain semidirect product p groups of constant nilpotency class and for the multi dimensional univariate hidden polynomial graph problem when the degree of the polynomials is constant 1

on spin systems related to the temperley lieb algebra the spectrum of the transfer matrices constructed from the spectral parameter dependent temperley lieb r matrix is found using functional relations identical to those of the spin 1 2 xxz magnet

epsilon net method for optimizations over separable states we give algorithms for the optimization problem max q where q is a hermitian matrix and the variable is a bipartite separable quantum state this problem lies at the heart of several problems in quantum computation and information such as the complexity of qma 2 while the problem is np hard our algorithms are better than brute force for several instances of interest in particular they give pspace upper bounds on promise problems admitting a qma 2 protocol in which the verifier performs only a logarithmic number of elementary gates on both proofs as well as the promise problem of deciding if a bipartite local hamiltonian has a large or small ground energy for q 0 our algorithm runs in time exponential in q f while the existence of such an algorithm was first proved recently by brandao christandl and yard 2011 8 our algorithm is conceptually simpler

efficient quantum algorithms for simulating sparse hamiltonians we present an efficient quantum algorithm for simulating the evolution of a quantum state for a sparse hamiltonian h over a given time t in terms of a procedure for computing the matrix entries of h in particular when h acts on n qubits has at most a constant number of nonzero entries in each row column and h is bounded by a constant we may select any positive integer k such that the simulation requires o log n t1 1 2k accesses to matrix entries of h we also show that the temporal scaling cannot be significantly improved beyond this because sublinear time scaling is not possible

boundary k matrices for the xyz xxz and xxx spin chains the general solutions for the factorization equations of the reflection matrices k or theta for the eight vertex and six vertex models xyz xxz and xxx chains are found the associated integrable magnetic hamiltonians are derived explicitly finding families depending on several continuous as well as discrete parameters

making classical ground state spin computing fault tolerant we examine a model of classical deterministic computing in which the ground state of the classical system is a spatial history of the computation this model is relevant to quantum dot cellular automata as well as to recent universal adiabatic quantum computing constructions in its most primitive form systems constructed in this model cannot compute in an error free manner when working at nonzero temperature however by exploiting a mapping between the partition function for this model and probabilistic classical circuits we are able to show that it is possible to make this model effectively error free we achieve this by using techniques in fault tolerant classical computing and the result is that the system can compute effectively error free if the temperature is below a critical temperature we further link this model to computational complexity and show that a certain problem concerning finite temperature classical spin systems is complete for the complexity class merlin arthur this provides an interesting connection between the physical behavior of certain many body spin systems and computational complexity

the kowalewski top 99 years later a lax pair generalizations and explicit solutions a natural lax pair for the kowalewski top is derived by using a general group theoretic approach this gives a new insight into the algebraic geometry of the top and leads to its complete solution via finite band integration theory

the symmetric group defies strong fourier sampling the dramatic exponential speedups of quantum algorithms over their best existing classical counterparts were ushered in by the technique of fourier sampling introduced by bernstein and vazirani and developed by simon and shor into an approach to the hidden subgroup problem this approach has proved successful for abelian groups leading to efficient algorithms for factoring extracting discrete logarithms and other number theoretic problems we show however that this method cannot resolve the hidden subgroup problem in the symmetric groups even in the weakest information theoretic sense in particular we show that the graph isomorphism problem cannot be solved by this approach our work implies that any quantum approach based upon the measurement of coset states must depart from the original framework by using entangled measurements on multiple coset states

on complexity of the quantum ising model we study complexity of several problems related to the transverse field ising model tim first we consider the problem of estimating the ground state energy known as the local hamiltonian problem lhp it is shown that the lhp for tim on degree 3 graphs is equivalent modulo polynomial reductions to the lhp for general k local stoquastic hamiltonians with any constant k ge 2 this result implies that estimating the ground state energy of tim on degree 3 graphs is a complete problem for the complexity class mathsf stoqma an extension of the classical class mathsf ma as a corollary we complete the complexity classification of 2 local hamiltonians with a fixed set of interactions proposed recently by cubitt and montanaro secondly we study quantum annealing algorithms for finding ground states of classical spin hamiltonians associated with hard optimization problems we prove that the quantum annealing with tim hamiltonians is equivalent modulo polynomial reductions to the quantum annealing with a certain subclass of k local stoquastic hamiltonians this subclass includes all hamiltonians representable as a sum of a k local diagonal hamiltonian and a 2 local stoquastic hamiltonian

separation of variables in the elliptic gaudin model for the elliptic gaudin model a degenerate case of the xyz integrable spin chain a separation of variables is constructed in the classical case the corresponding separated coordinates are obtained as the poles of a suitably normalized baker akhiezer function the classical results are generalized to the quantum case where the kernel of the separating integral operator is constructed the simplest one degree of freedom case is studied in detail

integrable quadratic classical hamiltonians on so 4 and so 3 1 we investigate a special class of quadratic hamiltonians on so 4 and so 3 1 and describe hamiltonians that have additional polynomial integrals one of the main results is a new integrable case with an integral of sixth degree