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

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

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

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

diffusion equation for multivalued stochastic differential equations abstract deterministic oscillations with bilinear hysteresis are governed by a multivalued differential equation of the type k b g where k is maximal monotonic and b is lipschitzian an existence and uniqueness result is proven for corresponding stochastic equation the diffusion equation satisfied by the laws of t is established in the particular case k 0 this equation is equivalent to the fokker planck equation

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

a posteriori error analysis for higher order dissipative methods for evolution problems we prove a posteriori error estimates for time discretizations by the discontinuous galerkin method dg q and the corresponding implicit runge kutta radau method irk r q of arbitrary order q 0 for both linear and nonlinear evolution problems of the form u prime mathfrak f u f with 2 angle bounded operator mathfrak f the key ingredient is a novel higher order reconstruction widehat u of the discrete solution u which restores continuity and leads to the differential equation widehat u prime pi mathfrak f u f for a suitable interpolation operator and piecewise polynomial approximation f of f we discuss applications to linear pde such as the convection diffusion equation 1 2 and the wave equation formally and nonlinear pde corresponding to subgradient operators 1 such as the p laplacian as well as lipschitz operators 1 2 we also derive conditional a posteriori error estimates for the time dependent minimal surface problem

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

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

l error estimate for an approximation of a parabolic variational inequality almost optimall convergence of an approximation of a variational inequality of parabolic type is proved under regularity assumptions which are met by the solution of a one phase stefan problem the discretization employs piecewise linear finite elements in space and the backward euler scheme in time by means of a maximum principle the problem is reduced to an error estimate for an auxiliary parabolic equation the latter bound is obtained by using the smoothing property of the galerkin method

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

convergence order of a numerical scheme for sweeping process in j venel numer math 118 2011 pp 367 400 an implementable algorithm was introduced to compute discrete solutions of sweeping processes i e specific first order differential inclusions the convergence of this numerical scheme was proved thanks to compactness arguments here we establish that this algorithm is of order frac 1 2 the considered sweeping process involves a set valued map given by a finite number of inequality constraints the proof rests on a metric qualification condition between the sets associated to each constraint

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

a note on green functors with inflation abstract this note is motivated by the problem to understand given a commutative ring f which g sets x y give rise to isomorphic f g representations f x f y a typical step in such investigations is an argument that uses induction theorems to give very general sufficient conditions for all such relations to come from proper subquotients of g in the present paper we axiomatise the situation and prove such a result in the generality of mackey functors and green functors with inflation our result includes as special cases a result of deligne on monomial relations a result of the first author and tim dokchitser on brauer relations in characteristic 0 and a new result on brauer relations in characteristic p 0 we will need the new result in a forthcoming paper on brauer relations in positive characteristic

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

numerical solution of the obstacle problem by the penalty method part ii time dependent problems in a previous paper we considered the possibility of using the penalty method to get approximations of the solution of elliptic variational inequalities in this note we show that the same idea can be used for the treatment of parabolic problems by coupling the penalty parameter e and the discretization parametersh and t quasi optimal error estimates are derived in suitable norms

algebraicisation of explicit brauer induction this paper deals with canonical forms for induction theorems in the complex finite dimensional representation theory of a finite group g the first example of an explicit form for an induction theorem is due to r brauer br who in 1951 gave a canonical form for artin s induction theorem in 1946 r brauer had also proved that each virtual character of g may be expressed as an integral linear combination of characters which are induced from linear i e one dimensional characters of subgroups of g one may reformulate brauer s induction theorem in the following manner define r g to be the free abelian group on the g conjugacy classes h 5 s of linear characters of subgroups define 6 r g r g by b h s indz rp brauer s theorem states that 6 is onto by an explicit brauer induction formula we will mean a section for the map 6 given such a section one may use it to obtain a presentation for r g in terms of generators h s c this problem is mentioned in ser p 71 footnote in 1986 snaith produced such a section t by means of the algebraic topology of smooth group actions related to g representations details appear in sl see also s2 s3 s4 in his thesis boltje algebraically constructed another such section a in fact it is possible to give a topological construction of a similar to that of sl for t this construction is given in 2 26 see theorem 2 27 in this paper we derive algebraic formulae for t and its topological ancestor ro in a manner similar to that used for a in bl these results appear in theorems 3 23 and 3 29 in section 4 we derive a formula which relates ac and t directly rather than by means of the algebraic formulae of section 3 the prerequisites for the understanding of a t and l the explicit

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

categorifying induction formulae via divergent series abstract we show how to get explicit induction formulae for finite group representations and more generally for rational green functors by summing a divergent series over dwyer s subgroup and centralizer decomposition spaces this results in formulae with rational coefficients the former space yields a well known induction formula the latter yields a new one as essentially immediate corollaries of the existing literature we get similar formulae in group cohomology and stable splittings of classifying spaces

difference methods for differential inclusions a survey the main objective of this survey is to study convergence properties of difference methods applied to differential inclusions it presents in a unified way a number of results scattered in the li

maximal monotone model with history term abstract models involving maximal monotone terms and history delay term are considered in a mathematical and numerical point of view mechanical example of elasto plastic material with creep is presented

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

study of a maximal monotone model with a delay term we present a model involving a friction term and a delay term by using results of existence and uniqueness for maximal monotone differential inclusions we give theoretical results for this model an implicit euler numerical scheme and results related to order of convergence are also provided from both a theoretical and a numerical point of view