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

characterization of sobolev and bv spaces abstract the main results of this paper are new characterizations of w 1 p 1 p and bv for r n an arbitrary open set using these results we answer some open questions of brezis 2002 10 and ponce 2004 25

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

a class of high order nonlocal operators we study a class of nonlocal operators that may be seen as high order generalizations of the well known nonlocal diffusion operators we present properties of the associated nonlocal functionals and nonlocal function spaces including nonlocal versions of sobolev inequalities such as the nonlocal poincare and nonlocal gagliardo nirenberg inequalities nonlocal characterizations of high order sobolev spaces in the spirit of bourgain brezis mironescu are provided applications of nonlocal calculus of variations to the well posedness of linear nonlocal models of elastic beams and plates are also considered

on a class of nonvariational problems in fractional orlicz sobolev spaces abstract in this paper we deal with a nonlinear problem driven by the fractional m laplacian and with a nonlinear nonhomogeneous reaction term first we give some further properties for the new fractional orlicz sobolev space then using the theory of nonlinear operators of monotone type we show the existence of a nonnegative solution our problem is nonvariational in nature

realization extension and classification of certain physically important groups and algebras an associative algebra of differential forms with division has been constructed the algebra of forms in each different space provides a practical realization of the universal clifford algebra of that space a classification of all such algebras is given in terms of two distinct types of algebras nk and sk the former include the dihedral quaternion and majorana algebras the latter include the complex spinor and dirac algebras the associative product expresses hodge duality as multiplication by a basis element this makes possible the realization of higher order algebras in a calculationally useful algebraic setting the fact that the associative algebras as well as the enveloped lie algebras are precisely those arising in physics suggests that this formalism may be a convenient setting for the formulation of basic physical laws

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

stability of solutions for nonlocal problems abstract in this paper we deal with the stability of solutions to fractional p laplace problems with nonlinear sources when the fractional parameter s goes to 1 we prove a general convergence result for general weak solutions which is applied to study the convergence of ground state solutions of p fractional problems in bounded and unbounded domains as s goes to 1 moreover our result applies to treat the stability of p fractional eigenvalues as s goes to 1

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

two subtle convex nonlocal approximations of the bv norm abstract inspired by the bbm formula and by work of g leoni and d spector we analyze the asymptotic behavior of two sequences of convex nonlocal functionals n u and n u which converge formally to the bv norm of u we show that pointwise convergence when u is not smooth can be delicate by contrast convergence to the bv norm is a robust and very useful mode of convergence

bourgain brezis mironescu domains abstract bourgain et al 2001 proved that for p 1 and smooth bounded domain r n lim s 1 1 s f x f y p x y n s p d x d y f x p d x for all f l p this gives a characterization of w 1 p by means of w s p seminorms only for the case p 1 davila 2002 proved that when is a bounded domain with lipschitz boundary lim s 1 1 s f x f y x y n s d x d y f b v for all f l 1 this characterizes b v in terms of w s 1 seminorm in this paper we extend the first result and partially extend the second result to extension domains

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

gamma convergence sobolev norms and bv functions we prove that the family of functionals i defined by i g rn rn g x g y x y n p dx dy g l r for p 1 and 0 converges in l r as goes to 0 when p 1 hereafter denotes the euclidean norm of r we also introduce a characterization for bv functions which has some advantages in comparison with the classic one based on the notion of essential variation on a e line

definable and contractible contracts this paper analyzes bayesian normal form games in which players write contracts that condition their actions on the contracts of other players these contracts are required to be representable in a formal language this is accomplished by constructing contracts which are definable functions of the godel code of every other player s contract we provide a complete characterization of the set of allocations supportable as pure strategy bayesian equilibria of this contracting game when information is complete this characterization provides a folk theorem in general the set of supportable allocations is smaller than the set supportable by a centralized mechanism designer

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

mechanism games with multiple principals and three or more agents we consider a class of mechanism games in which there are multiple principals and three or more agents for a mechanism game in this class a sort of folk theorem holds there is a threshold value for each of the principals such that an allocation is achieved at a pure strategy sequential equilibrium of the game if and only if i it is incentive compatible and ii it attains an expected utility for each principal that is greater than or equal to the threshold value for the principal

characterization of polynomials and higher order sobolev spaces in terms of functionals involving difference quotients abstract the aim of this paper which deals with a class of singular functionals involving difference quotients is twofold deriving suitable integral conditions under which a measurable function is polynomial and stating necessary and sufficient criteria for an integrable function to belong to a k th order sobolev space one of the main theorems is a new characterization of w k p k n and p 1 for arbitrary open sets r n in particular we provide natural generalizations of the results regarding sobolev spaces summarized in brezis overview article brezis 2002 to the higher order case and extend the work borghol 2007 to a more general setting

a constrained shape optimization problem in orlicz sobolev spaces abstract in this manuscript we study the following optimization problem given a bounded and regular domain r n we look for an optimal shape for the w vanishing window on the boundary with prescribed measure over all admissible profiles in the framework of the orlicz sobolev spaces associated to constant for the sobolev trace embedding in this direction we establish existence of minimizer profiles and optimal sets as well as we obtain further properties for such extremals finally we also place special emphasis on analyzing the corresponding optimization problem involving an a vanishing hole inside the domain with volume constraint

a minimum problem with free boundary in orlicz spaces abstract we consider the optimization problem of minimizing g u u 0 d x in the class of functions w 1 g with u 0 w 0 1 g for a given 0 0 and bounded w 1 g is the class of weakly differentiable functions with g u d x the conditions on the function g allow for a different behavior at 0 and at we prove that every solution u is locally lipschitz continuous that it is a solution to a free boundary problem and that the free boundary u 0 is a regular surface also we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the lipschitz regularity of the weak solutions and the c 1 regularity of their free boundaries near flat free boundary points

phase segregation dynamics in particle systems with long range interactions i macroscopic limits we present and discuss the derivation of a nonlinear nonlocal integrodifferential equation for the macroscopic time evolution of the conserved order parameter r t of a binary alloy undergoing phase segregation our model is ad dimensional lattice gas evolving via kawasaki exchange with respect to the gibbs measure for a hamiltonian which includes both short range local and long range nonlocal interactions the nonlocal part is given by a pair potential dj x y 0 x and y in d in the limit 0 the macroscopic evolution is observed on the spatial scale 1 and time scale 2 i e the density r t is the empirical average of the occupation numbers over a small macroscopic volume element centered atr x a rigorous derivation is presented in the case in which there is no local interaction in a subsequent paper part ii we discuss the phase segregation phenomena in the model in particular we argue that the phase boundary evolutions arising as sharp interface limits of the family of equations derived in this paper are the same as the ones obtained from the corresponding limits for the cahn hilliard equation

mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions in this paper we study the linear bond based nonlocal peridynamic models with a particular focus on problems associated with nonstandard nonlocal displacement loading conditions both stationary and time dependent problems are considered for a one dimensional scalar equation defined on a finite bar and for a two dimensional system defined on a square the related peridynamic operators and associated functional spaces are analyzed applications to the numerical analysis of the finite dimensional approximations to peridynamic models are also illustrated

algebras with three anticommuting elements i spinors and quaternions a general construction of alternative algebras with three anticommuting elements and a unit is given as an exhaustive result over the real and complex fields we obtain the clifford algebras h quaternions n1 dihedral clifford algebra which is related to real 2 spinors and s1 algebra of pauli matrices which is related to complex 2 spinors what is important is that the algebras n1 and s1 possess inverses everywhere except on a region akin to the light cone of the minkowski space while the quaternion algebra h has inverses everywhere except at the zero element we discuss the reasons why the three algebras n1 h and s1 are so difficult to distinguish in the representation space of 2 2 complex matrices

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

games with incomplete information played by bayesian players i iii this article originally appeared in management science november 1967 volume 14 number 3 pp 159 182 published by the institute of management sciences the paper develops a new theory for the analysis of games with incomplete information where the players are uncertain about some important parameters of the game situation such as the payoff functions the strategies available to various players the information other players have about the game etc however each player has a subjective probability distribution over the alternative possibilities in most of the paper it is assumed that these probability distributions entertained by the different players are mutually consistent in the sense that they can be regarded as conditional probability distributions derived from a certain basic probability distribution over the parameters unknown to the various players but later the theory is extended also to cases where the different players subjective probability distributions fail to satisfy this consistency assumption in cases where the consistency assumption holds the original game can be replaced by a game where nature first conducts a lottery in accordance with the basic probability distribution and the outcome of this lottery will decide which particular subgame will be played i e what the actual values of the relevant parameters will be in the game yet each player will receive only partial information about the outcome of the lottery and about the values of these parameters however every player will know the basic probability distribution governing the lottery thus technically the resulting game will be a game with complete information it is called the bayes equivalent of the original game part i of the paper describes the basic model and discusses various intuitive interpretations for the latter part ii shows that the nash equilibrium points of the bayes equivalent game yield bayesian equilibrium points for the original game finally part iii considers the main properties of the basic probability distribution

on the bourgain brezis and mironescu theorem concerning limiting embeddings of fractional sobolev spaces the article is concerned with the bourgain brezis and mironescu theorem on the asymptotic behaviour of the norm of the sobolev type embedding operator ws p lpn n sp as s 1 and s n p thei

exponentiation of the spinor representation of the lorentz group an exact finite expression for the exponentiation of the spin 12 spinor representation of the lorentz group is obtained from this expression an exact finite expression for the exponentiation of the vector representation of the lorentz group is derived the two expressions are compared with the literature in the special cases of either spatial rotations or boosts only

analysis and approximation of nonlocal diffusion problems with volume constraints a recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in mbrn the nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion including a notion of nonlocal flux the ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory in particular it is shown that fractional laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider the numerous applications elucidate different interpretations of the operator and the associated governing equations for example a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process sufficie

eigenvalues and minimizers for a non standard growth non local operator abstract in this article we study eigenvalues and minimizers of a fractional non standard growth problem we prove several properties on these quantities and their corresponding eigenfunctions

the lorentz group and the thomas precession ii exact results for the product of two boosts the product of two lorentz boosts in different directions is equal to the product of a pure boost and a spatial rotation to second order the resulting boost is simply the sum of the individual boosts and the rotation is responsible for the thomas precession here the resulting boost and the rotation correction are calculated exactly the calculation of higher order corrections to the sum of two finite lorentz boosts is not usually done and here previous exact results are critically compared the complete expression is to the best of the author s knowledge a new result

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

fractional order orlicz sobolev spaces abstract in this paper we define the fractional order orlicz sobolev spaces and prove its convergence to the classical orlicz sobolev spaces when the fractional parameter s 1 in the spirit of the celebrated result of bourgain brezis mironescu we then deduce some consequences such as convergence of the modulars and convergence of solutions for some fractional versions of the g operator as the fractional parameter s 1

recursion formulas for the lie integral abstract an explicit definition is given of a non linear integral which is termed the lie integral this integral is defined for lie algebra valued functions in terms of the representations of the algebra various properties of the integral are investigated and applications are given to differential equations applications to differential geometry will be given elsewhere

new characterizations of sobolev metric spaces abstract we provide new characterizations of sobolev ad bv spaces in doubling and poincare metric spaces in the spirit of the bourgain brezis mironescu and nguyen limit formulas holding in domains of r n