on overdetermined problems for a general class of nonlocal operators abstract we study the overdetermined problem for a large family of non local operators given by generators of subordinate brownian motions in particular this family includes the fractional laplacian relativistic stable operators etc we consider these problems in bounded domains exterior domains and in annular domains and we show that under suitable conditions the domains and solutions are both radially symmetric our method uses both analytic and probabilistic tools

density and tails of unimodal convolution semigroups abstract we give sharp bounds for the isotropic unimodal probability convolution semigroups when their levy khintchine exponent has matuszewska indices strictly between 0 and 2

least squares estimation of transformation parameters between two point patterns in many applications of computer vision the following problem is encountered two point patterns sets of points x sub i and x sub i i 1 2 n are given in m dimensional space and the similarity transformation parameters rotation translation and scaling that give the least mean squared error between these point patterns are needed recently k s arun et al 1987 and b k p horn et al 1987 presented a solution of this problem their solution however sometimes fails to give a correct rotation matrix and gives a reflection instead when the data is severely corrupted the proposed theorem is a strict solution of the problem and it always gives the correct transformation parameters even when the data is corrupted

products of involutions in o v abstract let v be a finite dimensional vector space over a commutative field of characteristic distinct from 2 let v carry a symmetric nondegenerate bilinear form the special orthogonal group is o v o v det 1 main result an element o v is a product of two involutions in o v ifand only if dim v 2 mod 4 or an orthogonal decomposition of v into orthogonally indecomposable modules contains a module of odd dimension

the analysis of golf swing as a kinematic chain using dual euler angle algorithm the manner in which anatomical rotation from an individual segment contributes to the position and velocity of the endpoint can be informative in the arena of many athletic events whose goals are to attain the maximal velocity of the most distal segment this study presents a new method of velocity analysis using dual euler angles and its application in studying rotational contribution from upper extremity segments to club head speed during a golf swing dual euler angle describes 3d movement as a series of ordered screw motions about each orthogonal axis in a streamlined matrix form the dual transformation matrix and allows the translation and rotation component to be described in the same moving frame applying this method in biomechanics is a novel idea and the authors have previously applied the methodology to clinical studies on its use in displacement analysis the focus of this paper is velocity analysis and applications in sports biomechanics in this study electrogoniometers biometrics uk with a frequency of 1000 hz were attached to a subject during the execution of the swing to obtain the joint angles throughout the motion the velocity of the club head was then analyzed using the dual velocity which specifies the velocity distribution of a rigid body in screw motion at any point in time as the dual vector the contributions of each segment to the club head velocity were also compared in order to evaluate this method the calculated position and velocity of the club head were compared to the values obtained from video image analysis the results indicated that there is good agreement between calculated values and video data suggesting the suitability of using the dual euler method in analyzing a kinematic chain motion

whole body inverse dynamics over a complete gait cycle based only on measured kinematics this paper presents a three dimensional 3d whole body multi segment model for inverse dynamics analysis over a complete gait cycle based only on measured kinematic data the sequence of inverse dynamics calculations differs significantly from the conventional application of inverse dynamics using force plate data a new validated smooth transition assumption was used to solve the indeterminacy problem in the double support phase kinematic data is required for all major body segments and hence a whole body gait measurement protocol is presented finally sensitivity analyses were conducted to evaluate the effects of digital filtering and body segment parameters on the accuracy of the prediction results the model gave reasonably good estimates of sagittal plane ground forces and moment however the estimates in the other planes were less good which we believe is largely due to their small magnitudes in comparison to the sagittal forces and moment the errors observed are most likely caused by errors in the kinematic data resulting from skin movement artefact and by errors in the estimated body segment parameters a digital filtering cut off frequency of 4 5 hz was found to produce the best results it was also shown that errors in the mass properties of body segments can play a crucial role with changes in properties sometimes having a disproportionate effect on the calculated ground reactions the implication of these results is that even when force plate data is available the estimated joint forces are likely to suffer from similar errors

expressing infinite matrices as products of involutions abstract we take into consideration ut k the group of upper triangular infinite matrices whose entries lying on the main diagonal are equal to either 1 or 1 we show that for any field k every element of this group can be expressed as a product of at most five involutions moreover we prove that if k is a field of characteristic different from 2 then four involutions suffice

regularity theory for fully nonlinear integro differential equations we consider nonlinear integro differential equations like the ones that arise from stochastic control problems with purely jump levy processes we obtain a nonlocal version of the abp estimate harnack inequality and interior c 1 regularity for general fully nonlinear integro differential equations our estimates remain uniform as the degree of the equation approaches two so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations

boundary regularity for fully nonlinear integro differential equations we study fine boundary regularity properties of solutions to fully nonlinear elliptic integro differential equations of order 2s with s 0 1 we consider the class of nonlocal operators l l 0 which consists of all the infinitesimal generators of stable levy processes belonging to the class l0 of caffarelli silvestre for fully nonlinear operators i elliptic with respec tt ol w e prove that solutions to iu f in u 0 in r n satisfy u d s c s for all 0 where d is the distance to a ndf l we expect the holder exponent s to be optimal or almost optimal for general right hand sides f l moreover we also expect the class l to be the largest scale invariant subclass of l0 for which this result is true in this direction we show that the class l0 is too large for all solutions to behave like d s the constants in all the estimates in this paper remain bounded as th eo rder of the equation approaches 2

estimation of the axis of a screw motion from noisy data a new method based on plucker lines the problems of estimating the motion and orientation parameters of a body segment from two n point set patterns are analyzed using the plucker coordinates of a line plucker lines the aim is to find algorithms less complex than those in conventional use and thus facilitating more accurate computation of the unknown parameters all conventional techniques use point transformation to calculate the screw axis in this paper we present a novel technique that directly estimates the axis of a screw motion as a plucker line the plucker line can be transformed via the dual number coordinate transformation matrix this method is compared with schwartz and rozumalski 2005 a new method for estimating joint parameters from motion data journal of biomechanics 38 107 116 in simulations of random measurement errors and systematic skin movements simulation results indicate that the methods based on plucker lines plucker line method are superior in terms of extremely good results in the determination of the screw axis direction and position as well as a concise derivation of mathematical statements this investigation yielded practical results which can be used to locate the axis of a screw motion in a noisy environment developing the dual transformation matrix dtm from noisy data and determining the screw axis from a given dtm is done in a manner analogous to that for handling simple rotations a more robust approach to solve for the dual vector associated with dtm is also addressed by using the eigenvector and the singular value decomposition

conjugacy classes of involutions in the lorentz group ω v and in so v abstract the lorentz group v is bireflectional and all involutions in v are conjugate more generally we give conditions for two involutions to be conjugate in so v provided that v is a vector space over a finite field or over an ordered field

classification of isometries of spaces of constant curvature and invariant subspaces abstract we study the varieties of invariant totally geodesic submanifolds of isometries of the spherical euclidean and hyperbolic spaces in each finite dimension we show that the dimensions of the connected components of these varieties determine the orbit type or the z class of the isometry for this purpose we introduce the segre symbol of an isometry a discrete invariant encoding the structure of its normal form which parametrizes z classes we then provide a description of the isomorphism type of the varieties of invariant subspaces in terms of the segre symbol

harnack inequalities for non local operators of variable order we consider harmonic functions with respect to the operator formula math under suitable conditions on n x h we establish a harnack inequality for functions that are nonnegative and harmonic in a domain the operator l is allowed to be anisotropic and of variable order

open sets of maximal dimension in complex hyperbolic quasi fuchsian space let pi 1 be the fundamental group of a closed surface sigma of genus g 1 one of the fundamental problems in complex hyperbolic geometry is to find all discrete faithful geometrically finite and purely loxodromic representations of pi 1 into su 2 1 the triple cover of the group of holomorphic isometries of h c 2 in particular given a discrete faithful geometrically finite and purely loxodromic representation rho 0 of pi 1 can we find an open neighbourhood of rho 0 comprising representations with these properties we show that this is indeed the case when rho 0 preserves a totally real lagrangian plane

harnack inequality for some classes of markov processes in this paper we establish a harnack inequality for nonnegative harmonic functions of some classes of markov processes with jumps

the punctured torus and lagrangian triangle groups in pu 2 1 we embed the teichmuller space of the once punctured torus t 1 1 into the set of conjugacy classes of groups generated by three anti holomorphic involutions i1 i2 and i3 lagrangian triangle groups acting on the complex hyperbolic plane hc we deform this embedding and obtain a three dimensional family e of discrete faithful and type preserving representations of the fundamental group of the once punctured torus ams classification 51m10 32m15 22e40

decomposition of complex hyperbolic isometries by involutions abstract a k reflection of the n dimensional complex hyperbolic space h c n is an element in u n 1 with negative type eigenvalue 1 of multiplicity k 1 and positive type eigenvalue 1 of multiplicity n k we prove that a holomorphic isometry of h c n is a product of at most four involutions and a complex k reflection k 2 along the way we prove that every element in su n is a product of four or five involutions according as n 2 mod 4 or n 2 mod 4 we also give a short proof of the well known result that every holomorphic isometry of h c n is a product of two anti holomorphic involutions

reversible quaternionic hyperbolic isometries abstract let g be a group an element g g is called reversible if it is conjugate to g 1 within g and called strongly reversible if it is conjugate to g 1 by an order two element of g let h h n be the n dimensional quaternionic hyperbolic space let psp n 1 be the isometry group of h h n in this paper we classify reversible and strongly reversible elements in sp n and sp n 1 also we prove that all the elements of psp n 1 are strongly reversible

regularity results for stable like operators for 1 2 we consider operators of the form lf x rd f x h f x 1 h 1 f x h a x h h d dh and for 0 1 we consider the same operator but where the f term is omitted we prove under appropriate conditions on a x h that any solution u to lu f will be in c if f c

the dirichlet problem for nonlocal operators with singular kernels convex and nonconvex domains abstract we study the interior regularity of solutions to the dirichlet problem l u g in u 0 in r n for anisotropic operators of fractional type l u x 0 d s n 1 d a 2 u x u x u x 1 2 s here a is any measure on s n 1 a prototype example for l is given by the sum of one dimensional fractional laplacians in fixed given directions when a c s n 1 and g is c solutions are known to be c inside but not up to the boundary however when a is a general measure or even when a is l s n 1 solutions are only known to be c 3 s inside we prove here that for general measures a solutions are c 1 3 s inside for all 0 whenever is convex when a l s n 1 we show that the same holds in all c 1 1 domains in particular solutions always possess a classical first derivative the assumptions on the domain are sharp since if the domain is not convex and the measure a is singular we construct an explicit counterexample for which u is not c 3 s for any 0 even if g and are c

commutator groups of orthogonal groups over the reals with emphasis on lorentz groups abstract let be the commutator subgroup of the n dimensional lorentz group we give a criterion when an element of is a product of 2 or 3 involutions of we prove that a real element of is 2 reflectional then we study orthogonal groups over the reals with arbitrary signature in this situation each real element in the kernel of the spinorial norm is 2 reflectional in the kernel of the spinorial norm a main result states that the commutator group p q of an orthogonal group o p q over the reals is 2 reflectional if and only if the signature p q satisfies p q p q 2 mod 4 for all special orthogonal groups over arbitrary fields we prove that real elements are 2 reflectional

bireflectionality of the weak orthogonal and the weak symplectic groups let v be a finite dimensional vector space q a quadratic form and fa f the bilinear form associated with q we assume v is regular then the orthogonal group o v is bireflectional i e every isometry in o v is a product of two involutory isometries in o v this has been shown in 8 if the field of scalars k has characteristic distinct from 2 and in 4 and 5 if char k 2 the latter papers also establish the bireflectionality for the symplectic group sp v again under the assumption that v f is regular and char k 2 for char k 2 the symplectic group is not bireflectional see 31 we shall extend the results just mentioned we shall drop the assumptions that v is finite dimensional and that v f is regular i e the vector space v may be infinite dimensional and the radical of v may be distinct from zero we shall use the notation and the concepts in 2 for every 7c e hom v v we define f n ue v m u and b n vz u zj e v the spaces f r and b n are called fix and path of z respectively the groups o v n e o v rad vc f z and dim b n co and sp v 71 e sp v rad v c f n and dim b n co are called the weak orthogonal and the weak symplectic group respectively

factoring formal maps into reversible or involutive factors abstract an element g of a group is called reversible if it is conjugate in the group to its inverse an element is an involution if it is equal to its inverse this paper is about factoring elements as products of reversibles in the group g n of formal maps of c n 0 i e formally invertible n tuples of formal power series in n variables with complex coefficients the case n 1 was already understood 25 each product f of reversibles has linear part l f of determinant 1 the main results are that for n 2 each map f with det l f 1 is the product of 2 3 ceiling log 2 n reversibles and may also be factored as the product of 9 6 ceiling log 2 n involutions where the ceiling of x is the smallest integer x

decomposable representations and lagrangian submanifolds of moduli spaces associated to surface groups the importance of explicit examples of lagrangian submanifolds of moduli spaces is revealed by papers such as dostoglou and salamon ann of math 2 139 3 581 640 1994 and salamon proceedings of the international congress of mathematicians vol 1 2 zurich 1994 pp 526 536 birkhauser basel 1995 given a 3 manifold m with boundary m dostoglou and salamon use such examples to obtain a proof of the atiyah floer conjecture relating the symplectic floer homology of the representation space hom 1 m u u associated to an explicit pair of lagrangian submanifolds of this representation space and the instanton homology of the 3 manifold m in the present paper we construct a lagrangian submanifold of the space of representations mathcal m g l hom mathcal c pi g l u u of the fundamental group g lof a punctured riemann surface g l into an arbitrary compact connected lie group u this lagrangian submanifold is obtained as the fixed point set of an anti symplectic involution hat beta defined on mathcal m g l we show that the involution hat beta is induced by a form reversing involution defined on the quasi hamiltonian space u times u g times mathcal c 1 times cdots times mathcal c l the fact that hat beta has a non empty fixed point set is a consequence of the real convexity theorem for group valued momentum maps proved in schaffhauser a real convexity theorem for quasi hamiltonian actions submitted 25 p 2007 http arxiv org abs math 0705 0858 the notion of decomposable representation provides a geometric interpretation of the lagrangian submanifold thus obtained

each symplectic matrix is a product of four symplectic involutions abstract gustafson halmos and radjavi in 1973 proved that each matrix a with det a 1 is a product of four involutions we prove that these involutions can be taken to be symplectic if a is symplectic every symplectic matrix has unit determinant using this result we give an alternative proof of laffey s theorem that every nonsingular even size matrix is a product of skew symmetric matrices ballantine in 1978 proved that each matrix a with det a 1 is a product of four coninvolutions we prove that these coninvolutions can be taken to be symplectic if a is symplectic we also prove that each hamiltonian matrix is a sum of two square zero hamiltonian matrices

every real symplectic matrix is a product of real symplectic involutions abstract a 2n by 2n matrix a is symplectic if a t 0 i i 0 a 0 i i 0 it is known that if n 1 then every 2n by 2n complex symplectic matrix is a product of four symplectic involutions we consider the real case we give an example of a real symplectic matrix which is a product of two complex symplectic involutions but is not a product of two real symplectic involutions we show that every 4 by 4 real symplectic matrix is a product of four real symplectic involutions we use this result to show that for n 1 every 2n by 2n real symplectic matrix is a product of a finite number of real symplectic involutions

use of dual euler angles to quantify the three dimensional joint motion and its application to the ankle joint complex this paper presents a modified euler angles method dual euler angles approach to describe general spatial human joint motions in dual euler angles approach the three dimensional joint motion is considered as three successive screw motions with respect to the axes of the moving segment coordinate system accordingly the screw motion displacements are represented by dual euler angles the algorithm for calculating dual euler angles from coordinates of markers on the moving segment is also provided in this study as an example the proposed method is applied to describe motions of ankle joint complex during dorsiflexion plantarflexion a flock of birds electromagnetic tracking device fob was used to measure joint motion in vivo preliminary accuracy tests on a gimbal structure demonstrate that the mean errors of dual euler angles evaluated by using source data from fob are less than 11 for rotations and 1 mm for translations respectively based on the pilot study fob is feasible for quantifying human joint motions using dual euler angles approach r 2002 elsevier science ltd all rights reserved

barriers exit time and survival probability for unimodal levy processes we give superharmonic functions and derive sharp bounds for the expected exit time and probability of survival for isotropic unimodal levy processes in smooth domains

on products of two involutions in the orthogonal group of a vector space let v be a finite dimentional vector space over a commutative field of characteristic distinct from 2 let v carry a symmetric nondegenerate bilinear form results a let where o v and are involutions there exists an orthogonal decomposition of v into orthogonally indecomposable modules which are simultaneously invariant under and b let o v one can find involutions o v such that and b b b holds if and only if an orthogonal decomposition of v into orthogonally indecomposable modules does not contain a term whose minimum polynomial is x 1 where is even

on bireflectionality and trireflectionality of orthogonal groups let v be a finite dimensional vector space over a field of characteristic different from two let v carry a symmetric bilinearform we show that the weak orthogonal group o v is always trireflectional and is bireflectional if and only if the index of the form or the dimension of the radical is at most 1

closed loop problems in biomechanics part ii an optimization approach abstract a closed loop problem in biomechanics may be defined as a problem in which there are one or more closed loops formed by the human body in contact with itself or with an external system under certain conditions the problem is indeterminate the unknown forces and torques outnumber the equations force transducing devices which would help solve this problem have serious drawbacks and existing methods are inaccurate and non general the purposes of the present paper are 1 to develop a general procedure for solving closed loop problems 2 to illustrate the application of the procedure and 3 to examine the validity of the procedure a mathematical optimization approach is applied to the solution of three different closed loop problems walking up stairs vertical jumping and cartwheeling the following conclusions are drawn 1 the method described is reasonably successful for predicting horizontal and vertical reaction forces at the distal segments although problems exist for predicting the points of application of these forces 2 the results provide some support for the notion that the human neuromuscular mechanism attempts to minimize the joint torques and thus to a certain degree the amount of muscular effort 3 in the validation procedure it is desirable to have a force device for each of the distal segments in contact with a fixed external system and 4 the method is sufficiently general to be applied to all classes of closed loop problems

boundary regularity for nonlocal operators with kernels of variable orders abstract we study the boundary regularity of solutions of the dirichlet problem for the nonlocal operator with a kernel of variable orders since the order of differentiability of the kernel is not represented by a single number we consider the generalized holder space we prove that there exists a unique viscosity solution of l u f in d u 0 in r n d where d is a bounded c 1 1 open set and that the solution u satisfies u c v d and u v d d c d with the uniform estimates where v is the renewal function and d d x dist x d

closed loop problems in biomechanics part i a classification system biomechanics researchers have relied heavily on the inverse dynamics approach for calculating the forces and torques at human joints however implicit in this approach is the assumption that there are sufficient independent equations of motion to uniquely determine these unknown kinetics there exists a class of problems commonly referred to as closed loop problems when there are insufficient equations and indeterminacy arises the purposes of the present paper are 1 to develop a general classification system of closed loop problems for whole body movements and 2 to identify the minimum number of force transducing devices necessary to uniquely determine joint kinetics for these problems the classification system is based on the human subject s interaction with his environment and with himself two criteria are considered first the number of the subject s extremities in contact with fixed external reference systems and second the number of closed loops formed by those extremities not in contact with fixed external systems different combinations of these two criteria are examined and grouped into five cases according to the degree to which the equations of motion are over determined determined or under determined examples are given to illustrate the concepts it is felt that the use of this system should aid in the understanding of joint force and torque calculations especially with regard to the under determined cases

the moduli space of the modular group in complex hyperbolic geometry we construct the space of discrete faithful type preserving representations of the modular group into the isometry group of complex hyperbolic 2 space up to conjugacy this is the first fuchsian group for which the entire complex hyperbolic deformation space has been constructed we also show how the spheres of falbel zocca are related to the spheres hybrid spheres of schwartz

dirichlet heat kernel estimates for rotationally symmetric levy processes in this paper we consider a large class of purely discontinuous rotationally symmetric levy processes we establish sharp two sided estimates for the transition densities of such processes killed upon leaving an open set d w hend is a fat open set the sharp two sided estimates are given in terms of surviving probabilities and the global transition density of the levy process when d is a c 1 1 open set and the levy exponent of the process is given by 2 with being a complete bernstein function satisfying a mild growth condition at infinity our two sided estimates are explicit in terms of the distance function to the boundary of d and the levy density of x this gives an affirmative answer to the conjecture posted in chen kim and song global heat kernel estimates for relativistic stable processes in half space like open sets potential anal 36 2012 235 261 our results are the first sharp two sided dirichlet heat kernel estimates for a large class of symmetric levy processes with general levy exponents we also derive an explicit lower bound estimate for symmetric levy processes on r d in terms of their levy exponents

reversible complex hyperbolic isometries let pu n 1 denote the isometry group of the n dimensional complex hyperbolic space hn an isometry g is called reversible if g is conjugate to g 1 in pu n 1 if g can be expressed as a product of two involutions it is called strongly reversible we classify reversible and strongly reversible elements in pu n 1 we also investigate reversibility and strong reversibility in su n 1

asymptotic behavior in degenerate parabolic fully nonlinear equations and its application to elliptic eigenvalue problems abstract we study the fully nonlinear parabolic equation f d 2 u m u t 0 in 0 m 1 with the dirichlet boundary condition and positive initial data in a smooth bounded domain r n provided that the operator f is uniformly elliptic and positively homogeneous of order one we prove that the renormalized limit of parabolic flow u x t as t is the corresponding positive eigenfunction which solves f d 2 p 0 in where 0 p 1 m 1 and 0 is the corresponding eigenvalue we also show that some geometric property of the positive initial data is preserved by the parabolic flow under the additional assumptions that is convex and f is concave as a consequence the positive eigenfunction has such geometric property that is log is concave in the case p 1 and 1 p 2 is concave for 0 p 1

a three dimensional kinematic method for determining the effectiveness of arm segment rotations in producing racquet head speed the contribution that a segment s anatomical rotations make to racquet head speed depends on both the segment s angular velocity and the instantaneous position of the head of the racquet with respect to the segment s axes of rotation any analysis of racquet swing technique that does not consider both of these factors simultaneously is at best incomplete with this in mind a three dimensional kinematic method was developed to determine the effectiveness of the anatomical rotations of the upper arm forearm and hand in producing racquet head speed the method entailed developing a system of vector equations for three dimensional upper limb rotations that used displacement histories of 10 selected landmarks as input the required three dimensional displacement histories were obtained using three cine cameras and the dlt approach to test the diagnostic capabilities of the method a tennis serve was selected for analysis for the player and serve analyzed the greatest contribution to racquet head speed at impact was produced by internal rotation of the upper arm 8 m s 1 forearm pronation although exhibiting the fastest rotation at impact 24 rad s 1 ranked only fourth in terms of its contribution 4 m s 1 to racquet head speed to test the performance of the method a comparison was made between the racquet head speed measured directly from film and the racquet head speed computed by summing all of the individual segment contributions to speed commencing at the start of forward swing and ending at ball contact the results indicate that the method can successfully determine the individual contributions that the different anatomical rotational velocities of the arm segments make to the measured instantaneous racquet head speed

fully degenerate monge ampere equations abstract in this paper we consider the following nonlinear eigenvalue problem for the monge ampere equation find a non negative weakly convex classical solution f satisfying det d 2 f f p in f on for a strictly convex smooth domain r 2 and 0 p 2 when f 0 contains a convex domain we find a classical solution which is smooth on f 0 and whose free boundary f 0 is also smooth