the algebra of complex 2 2 matrices and a general closed baker campbell hausdorff formula we derive a closed formula for the baker campbell hausdorff series expansion in the case of complex matrices for arbitrary matrices a and b and a matrix z such that our result expresses z as a linear combination of a and b their commutator and the identity matrix i the coefficients in this linear combination are functions of the traces and determinants of a and b and the trace of their product the derivation proceeds purely via algebraic manipulations of the given matrices and their products making use of relations developed here based on the cayley hamilton theorem as well as a characterization of the consequences of and or its determinant being zero or otherwise as a corollary of our main result we also derive a closed formula for the zassenhaus expansion we apply our results to several special cases most notably the parametrization of the product of two matrices and a verification of the recent result of van brunt and visser 2015 j phys a math theor 48 225207 for complex matrices in this latter case deriving also the related zassenhaus formula which turns out to be quite simple we then show that this simple formula should be valid for all matrices and operators

baker campbell hausdorff relation for special unitary groups su n multiplication of two elements of the special unitary group su n determines uniquely a third group element a baker campbell hausdorff relation is derived which expresses the group parameters of the product written as an exponential in terms of the parameters of the exponential factors this requires the eigenvalues of three matrices consequently the relation can be stated analytically up to n 4 in principle similarity transformations encoding the time evolution of quantum mechanical observables for example can be obtained by the same means

forecasting cause age specific mortality using two random processes mortality forecasts are critical information for assessing the health of a population and are necessary for making informed decisions about how best to direct health related resources and activities timeliness in making health statistics available is crucial to identify and address current health problems being motivated to meet these needs we propose a method to forecast the number of cause age specific deaths through a two random processes model unlike the previous methods the new method incorporates both cross sectional and longitudinal correlations into our model without a high dimensional problem a bootstrap confidence interval is presented to measure the validity of our model and to detect an unusual occurrence of deaths our data analysis demonstrates that our method gives promising results compared with the true final counts

special case closed form of the baker campbell hausdorff formula the baker campbell hausdorff formula is a general result for the quantity where x and y are not necessarily commuting for completely general commutation relations between x and y the free lie algebra the general result is somewhat unwieldy however in specific physics applications the commutator while non zero might often be relatively simple which sometimes leads to explicit closed form results we consider the special case and show that in this case the general result reduces to furthermore we explicitly evaluate the symmetric function demonstrating that and relate this to previously known results for instance this result includes but is considerably more general than results obtained from either the heisenberg commutator or the creation destruction commutator

bootstrap prediction intervals for autoregression abstract the nonparametric bootstrap is applied to the problem of prediction in autoregression let y t t 0 1 2 be a stationary autoregressive process of known order p ar p given a realization of the series up to time t y 1 y 2 y t a 100 prediction interval for y t k is desired standard forecasting techniques which assume that the error sequence of the process y t is gaussian rely on the fact that the conditional distribution of y t k given the data is gaussian as well as a nonparametric alternative the bootstrap provides an estimate of the conditional distribution of y t k the method is similar to other applications of the bootstrap for linear models because the residuals are resampled the proposed methodology represents a different approach since an alternative representation for ar p series is used allowing for bootstrap replicates generated backward in time it follows that the resulting replicates all have the same conditionally fixed values at the end

the foundations of finite sample estimation in stochastic processes generally in the literature on stochastic processes estimation is investigated in terms of asymptotic properties in this paper we establish some finite sample optimal estimation let y1 y2 be a discrete stochastic process since we will be discussing results for samples of finite size n we restrict the process to r in the following the term parameter is used in the same broad sense as by godambe thompson 1984 let y be a class of probability distributions f on r and 0 0 f f e jz be a real parameter let hi be a real function of y 1 yi and 0 be such that

generalized coherent states for su n systems generalized coherent states are developed for su n systems for arbitrary n this is done by first iteratively determining explicit representations for the su n coherent states and then determining parametric representations useful for applications for su n the set of coherent states is isomorphic to a coset space su n su n 1 and thus shows the geometrical structure of the coset space these results provide a convenient 2n 1 dimensional space for the description of arbitrary su n systems we further obtain the metric and measure on the coset space and show some properties of the su n coherent states

modeling hazard rates as functional data for the analysis of cohort lifetables and mortality forecasting as world populations age the analysis of demographic mortality data and demographic predictions of future mortality have met with increasing interest the study of mortality patterns and the forecasting of future mortality with its associated impacts on social welfare health care and societal planning has become a more pressing issue an ideal set of data to study patterns of change in long term mortality is the well known historical swedish cohort mortality data because of its high quality and long span of more than two centuries we explore the use of functional data analysis to model these data and to derive mortality forecasts specifically we address the challenge of flexibly modeling these data while including the effect of the birth year by regarding log hazard functions derived from observed cohort lifetables as random functions a functional model for the analysis of these cohort log hazard functions extending functional principal component approaches by introducing time varying eigenfunc

on the single exponential closed form of the product of two exponential operators in the frame of a vectorial pauli algebraic approach it is shown that the product of two exponentials of any two by two linear operators can be put in a single exponential closed form as a first application a compact vectorial expression for the characteristics angle and axis of the product of two r3 rotations is established the same mathematics can be used in a large diversity of problems of the whole class of two states physical systems an exemplification in the field of polarization optics is given

some explicit formulas for the matrix exponential formulas are derived for the exponential of an arbitrary 2 2 matrix in terms of either its eigenvalues or entries these results are then applied to the second order mechanical vibration equation with weak or strong damping some formulas for the exponential of n n matrices are given for matrices that satisfy an arbitrary quadratic polynomial besides the above mentioned 2 2 matrices these results encompass involutory rank 1 and idempotent matrices consideration is then given to n n matrices that satisfy a special cubic polynomial these results are applied to the case of a 3 3 skew symmetric matrix whose exponential represents the constant rotation of a rigid body about a fixed axis

the exponential mapping in clifford algebras the exponential mapping takes the lie algebra of the lorentz group into the lorentz group each element of the group is defined as a formal power series while the product of two exponential elements usually involves an infinite sum of commutator terms such as in the baker campbell hausdorff formula because of the special arithmetic in clifford algebras many baker campbell hausdorff like formulas and identities can be calculated or summed exactly when they involve only elements from the algebra we calculate exact identities for the baker campbell hausdorff formula and related formulas in the quaternion and dihedral algebras these are useful in treatments of the lorentz group and make possible a truly finite as opposed to infinitesimal description of a transformation group in physics

common hypercyclic functions for translation operators with large gaps abstract let h c be the set of entire functions endowed with the topology of local uniform convergence fix a sequence of non zero complex numbers n n which satisfies the following property for every m 0 there exists a subsequence n of n such that i n 1 n m for every n 1 2 and ii n 1 1 n we prove that there exists a residual set g h c such that for every f g and every non zero complex number a the set f z n a n 1 2 is dense in h c this answers in the affirmative question 1 in 29 and it also provides an extension of a theorem due to costakis and sambarino in 32

comment on special case closed form of the baker campbell hausdorff formula recently van brunt and visser 2015 j phys a math theor 48 225207 succeeded in explicitly evaluating the baker campbell hausdorff bch expansion series for the noncommuting operators x and y provided that the two operators satisfy the commutation relation and the operator i commutes with both of them in this comment we show that the closed form bch formula of this special case can be straightforwardly derived by the means of the wei norman theorem and no summation of the infinite series is needed

riemannian geometry of quantum computation abstract a review is given of some recent developments in the differential geometry of quantum computation for which the quantum evolution is described by the special unitary unimodular group s u 2 n using the lie algebra s u 2 n detailed derivations are given of a useful riemannian geometry of s u 2 n including the connection and the geodesic equation for minimal complexity quantum computations

some open problems in matrix theory arising in linear systems and control control theory has long provided a rich source of motivation for developments in matrix theory accordingly we discuss some open problems in matrix theory arising from theoretical and practical issues in linear systems theory and feedback control the problems discussed include robust stability matrix exponent induced norms stabilizability and pole assignability and nonstandard matrix equations a substantial number of references are included to acquaint matrix theorists with problems and trends in this application area feedback control theory has long provided a rich source of motivation for developments in matrix theory the purpose of this paper is to discuss several open problems in matrix theory that arise from theoretical and practical issues in feedback control theory and the associated area of linear systems theory many of these problems are remarkably simple to state are of intense interest in control theory and applications and yet remain unsolved besides leading to the resolution of these problems it is hoped that this paper will help to stimulate increased interaction between matrix and control theorists accord

coherent states and the classical limit on irreducible representations we give an explicit construction of the coherent states for an arbitrary irreducible representation we also construct the symplectic structure on the manifold of coherent states find canonical variables and discuss various classical limits of quantum mechanical systems with relevant observables that obey commutation relations

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

simplifying the reinsch algorithm for the baker campbell hausdorff series the goldberg version of the baker campbell hausdorff series computes the quantity z x y lnexey wg w w x y where x and y are not necessarily commuting in terms of words constructed from the x y alphabet the so called goldberg coefficients g w are the central topic of this article this baker campbell hausdorff series is a general purpose tool of very wide applicability in mathematical physics quantum physics and many other fields the reinsch algorithm for the truncated series permits one to calculate the goldberg coefficients up to some fixed word length w by using nilpotent w 1 w 1 matrices we shall show how to further simplify the reinsch algorithm making its implementation in principle utterly straightforward using off the shelf symbolic manipulation software specific computations provide examples which help to provide a deeper understanding of the goldberg coefficients and their properties for instance we shall establish some strict bounds and some equalities on the number of non zero goldberg coefficients unfortunately we shall see that the number of nonzero goldberg coefficients often grows very rapidly in fact exponentially with the word length w furthermore the simplified reinsch algorithm readily generalizes to many closely related but still quite distinct problems we shall also present closely related results for the symmetric product s x y lnex 2eyex 2 wgs w w x y variations on such themes are straightforward for instance one can just as easily consider the loop product l x y lnexeye xe y wgl w w x y this loop type of series is of interest for instance when considering either differential geometric parallel transport around a closed curve non abelian versions of stokes theorem or even wigner rotation thomas precession in special relativity several other closely related series are also briefly investigated

a baker campbell hausdorff disentanglement relation for lie groups a bch formula for groups having faithful matrix representations is derived by a decomposition of the general group element exp m into factors each factor being an exponential involving the infinitesimal generators of the group this is a generalization of the authors earlier work for su 3

special cases of a matrix exponential formula abstract some low dimension or low rank cases of a formula for a product of exponentials of matrices are studied using eigenvalue considerations counterexamples to certain natural conjectures involving a product of matrix exponentials are also given and comments relating to certain recently discovered exponential identities are included

exact baker campbell hausdorff formula for the contact heisenberg algebra in this work we introduce the contact heisenberg algebra which is the restriction of the jacobi algebra on contact manifolds to the linear and constant functions we give the exact expression of its corresponding baker campbell hausdorff formula we argue that this result is relevant to the quantization of contact systems

quantum phases of a qutrit we consider various approaches to treat the phases of a qutrit although it is possible to represent qutrits in a convenient geometrical manner by resorting to a generalization of the poincare sphere we argue that the appropriate way of dealing with this problem is through phase operators associated with the algebra su 3 the rather unusual properties of these phases are caused by the small dimension of the system and are explored in detail we also examine the positive operator valued measures that can describe the qutrit phase properties