on a spectral flow formula for the homological index abstract consider a selfadjoint unbounded operator d on a hilbert space h and a one parameter norm continuous family of selfadjoint bounded operators a t t r that converges in norm to asymptotes a at then under certain conditions 22 that include the assumption that the operators d t d a t t r all have discrete spectrum the spectral flow along the path d t can be shown to be equal to the index of t d t when the latter is an unbounded fredholm operator on l 2 r h in 16 an investigation of the index spectral flow question when the operators in the path may have some essential spectrum was started but under restrictive assumptions that rule out differential operators in general in 11 the question of what happens when the fredholm condition is dropped altogether was investigated in these circumstances the fredholm index is replaced by the witten index in this paper we take the investigation begun in 11 much further we show how to generalize a formula known from the setting of the l 2 index theorem to the non fredholm setting restricting back to the case of selfadjoint fredholm operators our formula extends the result of 22 in the sense of relaxing the discrete spectrum condition it also generalizes some other fredholm operator results of 21 16 11 that permit essential spectrum for the operators in the path our result may also apply however when the operators d t have essential spectrum equal to the whole real line our main theorem gives a trace formula relating the homological index of 7 to an integral formula that is known for a path of selfadjoint fredholms with compact resolvent and with unitarily equivalent endpoints to compute spectral flow our formula however applies to paths of selfadjoint non fredholm operators we interpret this as indicating there is a generalization of spectral flow to the non fredholm setting

relative index theorems and supersymmetric scattering theory we discuss supersymmetric scattering theory and employ krein s theory of spectral shift functions to investigate supersymmetric scattering systems this is the basis for the derivation of relative index theorems on some classes of open manifolds as an example we discuss the de rham complex for obstacles in n and asymptotically flat manifolds it is shown that the absolute or relative euler characteristic of an obstacle in n may be obtained from scattering data for the laplace operator on forms with absolute or relative boundary conditions respectively in the case of asymptotically flat manifolds we obtain the chern gauss bonnet theorem for thel2 euler characteristic

vacuum expectation values of products of chiral currents in 3 1 dimensions an algebraic rule is presented for computing expectation values of products of local nonabelian charge operators for fermions coupled to an external vector potential in 3 1 space time dimensions the vacuum expectation value of a product of four operators is closely related to a cyclic cocycle in noncommutative geometry of alain connes the relevant representation of the current is constructed using kirillov s method of coadjoint orbits

conservative integration of rigid body motion by quaternion parameters with implicit constraints summary an angular momentum and energy conserving time integration algorithm for rigid body rotation is formulated in terms of the quaternion parameters and the corresponding four component conjugate momentum vector via hamilton s equations the introduction of an extended mass matrix leads to a symmetric set of eight state space equations of motion the extra inertial parameter serves as a multiplier on the kinematic constraint and it is demonstrated that convergence characteristics are improved by selecting this parameter somewhat larger than the inertial moments external loads enter these equations via the set of momentum equations initially the normalization of the quaternion array is introduced via a lagrange multiplier however this lagrange multiplier can be expressed explicitly in terms of the gradient of the external load potential and elimination of the lagrange multiplier from the final format leaves only an explicit projection applied to the external load potential gradient an algorithm is developed by forming a finite increment of the hamiltonian this procedure identifies the proper selection of increments and mean values and leads to an algorithm with conservation of momentum and energy implementation conservation properties and accuracy of the algorithm are illustrated by examples with a flying box and a spinning top copyright 2012 john wiley sons ltd

index formulas for generalized wiener hopf operators and boson fermion correspondence in 2n dimensions the kernels of operators associated with special chiral gauge trans formations kinks in the 2 dimensional dirac theory are explicitly deter mined the result is used to obtain index formulas for fredholm operators corresponding to continuous chiral gauge transformations moreover the fock space quadratic forms corresponding to the kinks are proved to converge to the dirac field as the kink size goes to zero it is also shown that for n i 2 mod4 the majorana field can be reached in a similar fashion

simple and efficient integration of rigid rotations suitable for constraint solvers simple and efficient way of integrating rigid rotations is presented the algorithm is stable second order accurate and in its explicit version involves evaluation of only two exponential maps per time step the semi explicit version of the proposed scheme improves upon the long term stability while it retains the explicitness in the force evaluation the algebraic structure of both schemes makes them suitable forthe analysis of constrained multi body systems the explicit algorithm is specifically aimed at the analysis involving small incremental rotations where its modest computational cost becomes the major advantage the semi explicit scheme naturally broadens the scope of possible applications copyright 2009 john wiley sons ltd

explicit momentum conserving integrator for dynamics of rigid bodies approximating the midpoint lie algorithm we reformulate the midpoint lie algorithm which is implicit in the torque calculation to achieve explicitness in the torque evaluation this is effected by approximating the impulse imparted over the time step with discrete impulses delivered at either the beginning of the time step or at the end of the time step thus we obtain two related variants both of which are explicit in the torque calculation but only first order in the time step both variants are momentum conserving and both are symplectic consequently drawing on the properties of the composition of maps we introduce another algorithm that combines the two variants in a single time step the resulting algorithm is explicit momentum conserving symplectic and second order its accuracy is outstanding and consistently outperforms currently known implicit and explicit integrators

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

generalized particle statistics in two dimensions examples from the theory of free massive dirac field in the framework of algebraic quantum field theory we analyze the anomalous statistics exhibited by a class of automorphisms of the observable algebra of the two dimensional free massive dirac field constructed by fermionic gauge group methods the violation of haag duality the topological peculiarity of a two dimensional space time and the fact that unitary implementers do not lie in the global field algebra account for strange behaviour of statistics which is no longer an intrinsic property of sectors since automorphisms are not inner we exploit asymptotic abelianness of intertwiners in order to construct a braiding for a suitable c tensor subcategory of end fancyscript a we define two inequivalent classes of path connected bi asymptopias selecting only those sets of nets which yield a true generalized statistics operator

nilpotent lie algebras of maximal rank and of kac moody type e6 1 let be the kac moody algebra associated to the affine cartan matrix f 4 1 each nilpotent lie algebra of type f 4 1 is isomorphic to a quotient of the positive part of we determine the isomorphism classes of nilpotent lie algebras of type f 4 1

semi implicit reversible algorithms for rigid body rotational dynamics this paper presents two semi implicit algorithms based on splitting methodology for rigid body rotational dynamics the first algorithm is a variation of partitioned runge kutta prk methodology that can be formulated as a splitting method the second algorithm is akin to a multiple time stepping scheme and is based on modified crouch grossman mcg methodology which can also be expressed as a splitting algorithm these algorithms are second order accurate and time reversible however they are not poisson integrators i e non symplectic these algorithms conserve some of the first integrals of motion but some others are not conserved however the fluctuations in these invariants are bounded over exponentially long time intervals these algorithms exhibit excellent long term behavior because of their reversibility property and their approximate poisson structure preserving property the numerical results indicate that the proposed algorithms exhibit superior performance compared to some of the currently well known algorithms such as the simo wong algorithm newmark algorithm discrete moser veselov algorithm lewis simo algorithm and the liemid ea algorithm

nilpotent lie algebras of kac moody affine type abstract the nilpotent lie algebras are grouped into kac moody types simple affine hyperbolic we study the affine type and show that it can be reduced to the simple type concerning the discrete series whereas the continuous families require in addition the study of grassmanians the explicit list for the rank 2 illustrates the whole article

l2 index and the selberg trace formula abstract a method is developed for computing the l 2 index of a locally symmetric elliptic differential operator d acting on a locally symmetric manifold m g k with g semisimple of real rank one and of finite co volume based on applying the selberg trace formula to the difference of the two heat kernels associated to d the applications include an extension of the osborne warner multiplicity formula to certain non integrable discrete series derived from the l 2 spinor formula and showing the existence in some cases of non invariant l 2 cohomology classes in the middle dimension via the l 2 signature formula

a numerical test of long time stability for rigid body integrators summary in the context of hamiltonian odes a necessary condition for an integrator to be symplectic or conjugate symplectic is that it nearly preserves the exact hamiltonian this paper introduces a numerical test of this necessity for rigid body methods it turns out that several rigid body integrators proposed in literature fail this test hence these integrators should be used with caution for long time simulation copyright 2011 john wiley sons ltd

rigid body dynamics in terms of quaternions hamiltonian formulation and conserving numerical integration in the present paper unit quaternions are used to describe the rotational motion of a rigid body the unit length constraint is enforced explicitly by means of an algebraic constraint correspondingly the equations of motion assume the form of differential algebraic equations daes a new route to the derivation of the mass matrix associated with the quaternion formulation is presented in contrast to previous works the newly proposed approach yields a non singular mass matrix consequently the passage to the hamiltonian framework is made possible without the need to introduce undetermined inertia terms the hamiltonian form of the daes along with the notion of a discrete derivative make possible the design of a new quaternion based energy momentum scheme two numerical examples demonstrate the performance of the newly developed method in this connection comparison is made with a quaternion based variational integrator a director based energy momentum scheme and a momentum conserving scheme relying on the discretization of the classical euler s equations

the index formula and the spectral shift function for relatively trace class perturbations we compute the fredholm index index da of the operator da d dt a on l2 r h associated with the operator path a t t where af t a t f t for a e t r and appropriate f l2 r h via the spectral shift function a a associated with the pair a a of asymptotic operators a a on the separable complex hilbert space h in the case when a t is generally an unbounded relatively trace class perturbation of the unbounded self adjoint operator a we derive a formula an extension of a formula due to pushnitski relating the spectral shift function a a for the pair a a and the corresponding spectral shift function h2 h1 for the pair of operators h2 h1 dada da da in this relative trace class context h2 h1 1 1 2 1 2 a a d 2 1 2for a e 0 this formula is then used to identify the fredholm index of da with 0 a a in addition we prove that index da coincides with the spectral flow spflow a t t of the family a t t r and also relate it to the fredholm perturbation determinant for the pair a a index da spflow a t t 0 a a 1lime 0im ln deth a iei a iei 1 0 h2 h1 with the choice of the branch of ln deth on c such that limim z ln deth a zi a zi 1 0 we also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta invariant

fermion current algebras and schwinger terms in 3 1 dimensions we discuss the restricted linear group in infinite dimensions modeled by the schatten class of rank 2p 4 which contains the 3 1 dimensional analogs of the loop groups and is closely related to yang mills theory with fermions in 3 1 dimensions we give an alternative to the construction of the highest weight representation of this group found by mickelsson and rajeev our approach is close to quantum field theory with the elements of this group regarded as bogoliubov transformations for fermions in an external yang mills field though these cannot be unitarily implemented in the physically relevant representation of the fermion field algebra we argue that they can be implemented by sesquilinear forms and that there is a regularized product of forms providing an appropriate group structure on the lie algebra level this gives an explicit non perturbative construction of fermion current algebras in 3 1 space time dimensions which explicity shows that the wave function renormalization required for a consistent definition of the currents and their lie bracket naturally leads to the schwinger term identical with the mickelsson rajeev cocycle though the explicit form of the schwinger term is given only for the casep 2 our arguments apply also to the restricted linear groups modeled by schatten classes of rank 2p 6 8 corresponding to current algebras in d 1 dimensions d 5 7

n 2 supersymmetric quantum mechanics on riemann surfaces with meromorphic superpotentials we constructn 2 supersymmetric quantum hamiltonians with meromorphic superpotentials on compact riemann surfaces and investigate the topological properties of these hamiltonians l2 cohomology groups for supercharge a deformed bar partial operator are considered and the witten index for the supersymmetric hamiltonian with meromorphic superpotential is calculated in terms of euler characteristic of the riemann surface and the degree of a divisor of poles for the differential of the superpotential

stability in yang mills theories at a solution of the yang mills equations ons4 or the yang mills higgs equation on 3 the hessian of the action functional defines a natural second order elliptic operator the number of negative eigenvalues of this operator is bounded below by a multiple of the relevant topological charge the proof of this assertion requires a relative index theorem for dirac type operators on n n 3

lie group variational integrators for the full body problem we develop the equations of motion for full body models that describe the dynamics of rigid bodies acting under their mutual gravity the equations are derived using a variational approach where variations are defined on the lie group of rigid body configurations both continuous equations of motion and variational integrators are developed in lagrangian and hamiltonian forms and the reduction from the inertial frame to a relative frame is also carried out the lie group variational integrators are shown to be symplectic to preserve conserved quantities and to guarantee exact evolution on the configuration space one of these variational integrators is used to simulate the dynamics of two rigid dumbbell bodies

explicit newmark verlet algorithm for time integration of the rotational dynamics of rigid bodies we reformulate the traditional velocity based vector space newmark algorithm for the rotational dynamics of rigid bodies that is for the setting of the so 3 lie group we show that the most naive re write of the vector space algorithm possesses the properties of symplecticity and almost momentum conservation thus we obtain an explicit algorithm for rigid body dynamics that matches or exceeds performance of existing algorithms but which curiously does not seem to have been considered in the open literature so far copyright 2005 john wiley sons ltd

the discrete null space method for the energy consistent integration of constrained mechanical systems part ii multibody dynamics summary in the present work rigid bodies and multibody systems are regarded as constrained mechanical systems at the outset the constraints may be divided into two classes i internal constraints which are intimately connected with the assumption of rigidity of the bodies and ii external constraints related to the presence of joints in a multibody framework concerning external constraints lower kinematic pairs such as revolute and prismatic pairs are treated in detail both internal and external constraints are dealt with on an equal footing the present approach thus circumvents the use of rotational variables throughout the whole time discretization after the discretization has been completed a size reduction of the discrete system is performed by eliminating the constraint forces in the wake of the size reduction potential conditioning problems are eliminated the newly proposed methodology facilitates the design of energy momentum methods for multibody dynamics the numerical examples deal with a gyro top cylindrical and planar pairs and a six body linkage copyright 2006 john wiley sons ltd

discrete routh reduction this paper develops the theory of abelian routh reduction for discrete mechanical systems and applies it to the variational integration of mechanical systems with abelian symmetry the reduction of variational runge kutta discretizations is considered as well as the extent to which symmetry reduction and discretization commute these reduced methods allow the direct simulation of dynamical features such as relative equilibria and relative periodic orbits that can be obscured or difficult to identify in the unreduced dynamics the methods are demonstrated for the dynamics of an earth orbiting satellite with a non spherical j2 correction as well as the double spherical pendulum the j2 problem is interesting because in the unreduced picture geometric phases inherent in the model and those due to numerical discretization can be hard to distinguish but this issue does not appear in the reduced algorithm where one can directly observe interesting dynamical structures in the reduced phase space the cotangent bundle of shape space in which the geometric phases have been removed the main feature of the double spherical pendulum example is that it has a non trivial magnetic term in its reduced symplectic form our method is still efficient as it can directly handle the essential non canonical nature of the symplectic structure in contrast a traditional symplectic method for canonical systems could require repeated coordinate changes if one is evoking darboux theorem to transform the symplectic structure into canonical form thereby incurring additional computational cost our method allows one to design reduced symplectic integrators in a natural way despite the non canonical nature of the symplectic structure

unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum we show that for rigid body dynamics the mid point rule formulated in body co ordinates exactly conserves energy and the norm of the angular momentum for incremental force free motions but fails to conserve the direction of the angular momentum vector further we show that the mid point rule formulated in the spatial representation is in general physically and geometrically meaningless an alternative algorithm is developed which exactly preserves energy and the total spatial angular momentum in incremental force free motions the implicit version of this algorithm is unconditionally stable and second order accurate the explicit version conserves exactly angular momentum in incremental force free motions numerical simulations are presented which illustrate the excellent performance of the proposed procedure even for incremental rotations over 65 degrees the procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells