a representation theoretic approach to the dft with noncommutative generalizations it is known that both the one dimensional and multidimensional dfts discrete fourier transforms can be constructed as transition matrices associated with the decomposition of finite dimensional complex commutative group algebras into simple components two key attributes of these transforms orthogonality and the convolution property are inherent in such a description suggesting the possibility of enlarging the class by extending the construction to noncommutative groups in this context one speaks of a noncommutative or generalized transform the definition of which is based on the theory of semisimple rings the author reviews the ring theory and representation theory fundamental to the existence and computation of group algebra decompositions and sketches the representation theoretic construction of both the classical and noncommutative discrete fourier transforms the noncommutative transform associated with the class of dihedral groups is explicitly constructed and shown directly to exhibit both orthogonality and a noncommutative convolution property

kinetic analysis of the lower limbs during walking what information can be gained from a three dimensional model kinetic analyses joint moments powers and work of the lower limbs were performed during normal walking to determine what further information can be gained from a three dimensional model over planar models it was to be determined whether characteristic moment and power profiles exist in the frontal and transverse planes across subjects and how much work was performed in these planes kinetic profiles from nine subjects were derived using a three dimensional inverse dynamics model of the lower limbs and power profiles were then calculated by a dot product of the angular velocities and joint moments resolved in a global reference system characteristic joint moment profiles across subjects were found for the hip knee and ankle joints in all planes except for the ankle frontal moment as expected the major portion of work was performed in the plane of progression since the goal of locomotion is to support the body against gravity while generating movements which propel the body forward however the results also showed that substantial work was done in the frontal plane by the hip during walking 23 of the total work at that joint the characteristic joint profiles suggest defined motor patterns and functional roles in the frontal and transverse planes kinetic analysis in three dimensions is necessary particularly if the hip joint is being examined as a substantial amount of work was done in the frontal plane of the hip to control the pelvis and trunk against gravitational forces

the redundant nature of locomotor optimization laws a sagittal plane model of the lower limb which considered the possibility of antagonistic and synergistic muscle action and took account of the load bearing roles of the cruciate ligaments was applied to a dynamic analysis of level walking it was hypothesized that 1 the simple one sided constraints that intra articular contact forces must be compressive and muscle and ligament forces tensile substantially reduce the redundancy of the load transmitting structures of the lower limb 2 many previously proposed optimization laws for muscle selection yield equivalent results when they are applied to a finite set of admissible limiting solutions and 3 the aforementioned optimization laws when applied to a finite set of admissible limiting solutions do not adequately predict the co contraction of antagonistic muscles during gait the problem of indeterminacy was resolved by considering all possible limiting solutions of the system unknowns on the dynamic equations although 498 limiting solutions of nine unknowns could arise at each sampled point on the gait cycle the aforementioned one sided constraints ruled out the large majority of them it was shown that of the 498 possible the minimum number of simultaneous admissible solutions for any subject was as few as three and the maximum number was only 18 the principles of minimal total muscle force squared muscle force muscle stress intra articular contact force and instantaneous muscle power predicted remarkably similar patterns of muscle activity over the gait cycle of the six tested performance criteria the principle of minimal total ligament force was the least successful in terms of selecting solutions that closely matched the emg patterns this result implied that the muscles do not always act to protect the knee ligaments during gait finally each of the above minimum principles failed to predict any antagonistic quadriceps hamstrings action at the knee and hip around the event of heelstrike although such activity was indicated by electromyography

a wreath product group approach to signal and image processing ii convolution correlation and applications for pt i see ibid vol 48 no 1 p 102 32 2000 this paper continues the investigation of the use of spectral analysis on certain noncommutative finite groups wreath product groups in digital signal processing we describe the generalization of discrete cyclic convolution in convolution over these groups and show how it reduces to multiplication in the spectral domain finite group based convolution is defined in both the spatial and spectral domains and its properties established we pay particular attention to wreath product cyclic groups and further describe convolution properties from a geometric view point in terms of operations with specific signals and filters group based correlation is defined in a natural way and its properties follow from those of convolution the detection of similarity of perceptually similar signals and an application of correlation the detection of similarity of group transformed signals several examples using images are included to demonstrate the ideas pictorially

algebraic algorithms for sampling from conditional distributions we construct markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic examples include contingency tables logistic regression and spectral analysis of permutation data the algorithms involve computations in polynomial rings using grobner bases

muscular synergism ii a minimum fatigue criterion for load sharing between synergistic muscles a new physiological criterion for muscular load sharing is developed the criterion is based on the assumption that the endurance time of muscular contractions is maximized hence muscular fatigue is minimized the optimization problem is cast in the form of a linearly constrained non linear minimax optimization the new method predicts that 1 there is synergistic muscle action 2 muscle force increases non linearly with external force load 3 relatively more force is allocated to muscles that have a large maximum force large muscles 4 relatively more force is allocated to muscles with a high percentage of slow twitch fibers muscles that are fatigue resistant 5 the load sharing does not depend on the moment arm of the muscles although the absolute force levels do depend on this variable the predicted load sharing between two cat muscles during standing and walking is in good agreement with direct force measurement data from the literature

a non invasive protocol to determine the personalized moment arms of knee and ankle muscles one difficulty that comes with predicting muscular forces is the accuracy of experimental data particularly the assessment of muscle moment arms with respect to each joint rotation axis this paper presents a non invasive experimental protocol to obtain the personalized muscle moment arms with respect to the ankle and knee joints a specific pointer is used by a specialist of lower limb anatomy in order to define the local portion of the line of action of the different muscles closed to the joint on the standing subject s lower limb with this pointer the three dimensional coordinates of several points representing the line of action of 12 ankle and knee muscles are collected by a motion analysis system the collection is done five times by the same operator and one time by two different operators from this data the intra and inter operator repeatabilities are tested relative icc and absolute sem reliabilities are determined in order to evaluate the intra operator repeatability of this non invasive protocol the icc values obtained are higher than 0 91 for 10 among 12 muscles the intra operator repeatability is thus confirmed from the records realized by the two operators the differences are negligible thus the inter operator repeatability is also confirmed the moments arms obtained using this non invasive experimental protocol are compared with those calculated from origin and insertion points reported in the literature according to the work of whites pierrynowskis and kepples respectively the estimations obtained using the non invasive experimental protocol are found for some muscles more realistic than those calculated using the literature data and are always coherent with the role of the muscles described in anatomical books

a wreath product group approach to signal and image processing i multiresolution analysis we propose the use of spectral analysis on certain noncommutative finite groups in digital signal processing and in particular image processing we pay significant attention to groups constructed as wreath products of cyclic groups within this large class of groups our approach recovers the discrete fourier transform dft the haar wavelet transform various multichannel pyramid filter banks and other aspects of multiresolution analysis as special cases of a more general phenomenon in addition the group structure provides a rich algebraic structure that can be exploited for the analysis and manipulation of signals our approach relies on a synthesis of ideas found in the early work of holmes 1987 1990 karpovsky and trachtenberg 1985 and others on noncommutative filtering as well as diaconis s 1989 spectral analysis approach to understanding data

the three dimensional determination of internal loads in the lower extremity a three dimensional model of the lower limb containing 47 muscles was developed to study the differences between a two and three dimensional approach for determining internal loads the role of the dynamic joint representation and the behavior of different load bearing criteria in walking and running the problem of redundancy of the musculo skeletal system was resolved by applying inverse dynamics and static optimization methods different hypothetical load bearing capabilities of hinge spherical and intermediate joint types for the knee and the ankle joints were tested it was found that even almost planar movements such as walking and running are associated with significant three dimensional intersegment moments especially in the frontal plane thus a two dimensional approach may underestimate internal loads up to 60 it is shown that pure hinge joints are inappropriate for modeling the dynamical joint function of the knee and ankle joints a more flexible joint representation in combination with a squared muscle stress minimization criterion predicted a lot of synergistic as well as antagonistic muscle activation which was also found in the emg patterns the results indicate the importance of muscular joint stabilization in natural human movements compared to in vivo measurements it is speculated that the predicted force magnitudes are considerably overestimated due to error propagation and still insufficient anatomical models thus increased efforts to improve further the reliability of internal load calculations should be made in the future

effect of foot progression angle on hip joint moments during gait loosening has emerged as the most serious long term complication of total hip replacement and torsional loading of the femoral implant has been implicated as a possible cause in an effort to explore a strategy for minimizing this risk the following hypothesis was tested the foot progression angle has a significant effect on the resultant hip moments and particularly the internal external rotation moment during level walking twelve normal subjects performed a total of nine trials three in which they walked normally with the right foot pointing approximately straight ahead three in which they were told to walk with the foot internally rotated approximately 30 degrees and three in which the foot was externally rotated about 30 degrees the inverse dynamics approach was used to integrate the body segment parameter kinematic and force plate data and to solve for the resultant moment at the right hip joint in all three conditions foot straight foot in and foot out the subjects walked at the same average speed of 1 5 0 3 ms 1 for the flexion extension moment where the maximum flexion moment of 95 nm was in good agreement with other published data there was no significant difference between the three foot orientation conditions for the abduction adduction moment the foot straight condition exhibited the classic double peak pattern with a maximum abductor moment of 57 nm and there was no statistically significant difference between the three curves abstract truncated at 250 words

validation of optimization models that estimate the forces exerted by synergistic muscles the purpose of this study was to validate the mathematical predictions of individual muscle forces obtained using optimization models mathematical muscle force predictions were made using linear and nonlinear optimization models proposed in the literature actual muscle forces were measured experimentally from soleus gastrocnemius and plantaris muscles of an adult male cat during a variety of locomotor tasks mathematically predicted muscle forces did not agree well with the experimentally determined muscle forces because changes in force sharing during given step cycles and for different locomotor speeds were largely ignored in the theoretical models tested it is suggested that changes in force sharing during given step cycles may be caused by differences in force velocity characteristics or small time delays of onset of activation between muscles whereas changes in force sharing associated with different speeds of locomotion may be caused by corresponding changes in the magnitude of centrally controlled activation theoretical simulations testing the feasibility of these suggestions were encouraging in view of the findings of this study it is suggested that future models should be developed and validated based on experimental findings

a signal processing approach to fourier analysis of ranking data the importance of phase ranking data is a type of data obtained in some elections in customer surveys as well as from web search results such data may be considered as a type of signal defined on the group of permutations of n objects denoted sn there exists a fourier transform for sn obtained from group representation theory which is well known in the mathematics literature however previous work has not approached the transform from a signal processing perspective in particular there is no discussion of what constitutes magnitude and phase nor any analysis of what phase information might tell us beyond a well known connection to group translation this paper explores the properties of the phase spectrum of ranking data in particular a novel contribution is the formulation of the bispectrum for ranking data which may be used for studying phase linearity analysis of two well known ranking data sets shows that they are surprisingly well fit by linear phase approximations

frenet algorithm for simulations of fluctuating continuous elastic filaments we present an algorithm for generating the equilibrium configurations of fluctuating continuous elastic filaments based on a combination of statistical mechanics and differential geometry we use this to calculate the distribution function of the end to end distance of filaments with nonvanishing spontaneous curvature and show that for small twist and large bending rigidities there is an intermediate temperature range in which the filament becomes nearly completely stretched we show that volume interactions can be incorporated into our algorithm demonstrating this through the calculation of the effect of excluded volume on the end to end distance of the filament

comparison of global and joint to joint methods for estimating the hip joint load and the muscle forces during walking a three dimensional musculoskeletal model of the lower limb was developed to study the influence of biarticular muscles on the muscle force distribution and joint loads during walking a complete walking cycle was recorded for 9 healthy subjects using the standard optoelectronic motion tracking system ground contact forces were also measured using a 6 axes force plate inverse dynamics was used to compute net joint reactions forces and torques in the lower limb a static optimization method was then used to estimate muscle forces two different approaches were used in the first one named global method the biarticular muscles exerted a torque on the two joints they spanned at the same time and in the second one called joint by joint method these biarticular muscles were divided into two mono articular muscles with geometrical insertion origin via points and physiological properties remained unchanged the hip joint load during the gait cycle was then calculated taking into account the effect of muscle contractions the two approaches resulted in different muscle force repartition the biarticular muscles were favoured over any set of single joint muscles with the same physiological function when using the global method while the two approaches yielded only little difference in the resultant hip load the examination of muscle power showed that biarticular muscles could produce positive work at one joint and negative work at the other transferring energy between body segments and thus decreasing the metabolic cost of movement

fluctuating elastic rings statics and dynamics we study the effects of thermal fluctuations on elastic rings analytical expressions are derived for correlation functions of euler angles mean square distance between points on the ring contour radius of gyration and probability distribution of writhe fluctuations since fluctuation amplitudes diverge in the limit of vanishing twist rigidity twist elasticity is essential for the description of fluctuating rings we discover a crossover from a small scale regime in which the filament behaves as a straight rod to a large scale regime in which spontaneous curvature is important and twist rigidity affects the spatial configurations of the ring the fluctuation dissipation relation between correlation functions of euler angles and response functions is used to study the deformation of the ring by external forces the effects of inertia and dissipation on the relaxation of temporal correlations of writhe fluctuations are analyzed using langevin dynamics

conformations of closed dna we examine the conformations of a model for a short segment of closed dna the molecule is represented as a cylindrically symmetric elastic rod with a constraint corresponding to a specification of the linking number we obtain analytic expressions leading to the spatial configuration of a family of solutions representing distortions that interpolate between the circular form of dna and a figure eight form that represents the onset of interwinding we are also able to generate knotted loops we suggest ways to use our approach to produce other configurations relevant to studies of dna structure the stability of the distorted configurations is assessed along with the effects of fluctuations on the free energy of the various configurations

a physiologically based criterion of muscle force prediction in locomotion abstract the inversely nonlinear relationship of muscle contraction force and the possible contraction duration i e endurance is utilized in a method to mathematically predict individual muscle forces the method uses a nonlinear optimization scheme to solve the redundant distribution problem at the joints of interest the method is demonstrated at the elbow during isometric contraction and in the lower extremity during locomotion during gait the observed muscle activity pattern in the lower extremity as determined by emg shows substantial agreement with that activity pattern predicted when endurance is used as the optimization criterion the importance of selecting muscle prediction criteria based on sound physiological bases rather than on an arbitrary or mathematically convenient basis is emphasized

variation of muscle moment arms with elbow and forearm position we hypothesized that the moment arms of muscles crossing the elbow vary substantially with forearm and elbow position and that these variations could be represented using a three dimensional computer model flexion extension and pronation supination moment arms of the brachioradialis biceps brachialis pronator teres and triceps were calculated from measurements of tendon displacement and joint angle in two anatomic specimens and were estimated using a computer model of the elbow joint the anatomical measurements revealed that the flexion extension moment arms varied by at least 30 over a 95 degrees range of motion the changes in flexion extension moment arm magnitudes with elbow flexion angle were represented well by the computer model the anatomical studies and the computer model demonstrate that the biceps flexion moment arm peaks in a more extended elbow position and has a larger peak when the forearm is supinated also the peak biceps supination moment arm decreases as the elbow is extended these results emphasize the need to account for the variation of muscle moment arms with elbow flexion and forearm position

on the asymmetric elasticity of twisted dsdna abstract we present an analytical based proof for the consistency of two asymmetric elastic models of the double stranded dna in the regime around the b form we derive the shape equations in terms of euler angles for a uniform ribbon with noncircular cross section and spontaneous torsion we study the conditions to form a helical ribbon for the model and find that the euler angle must be either 0 or 2 we find that the helix is always stable or at least metastable for the model and there is either a continuous or discontinuous transition of extension from a twisted straight ribbon to a helical ribbon an over collapse phenomena of extension is observed for the ribbon under twisting the stability of the helix and the intrinsic asymmetric elasticity under twisting are also studied

fourier theoretic probabilistic inference over permutations permutations are ubiquitous in many real world problems such as voting ranking and data association representing uncertainty over permutations is challenging since there are n possibilities and typical compact and factorized probability distribution representations such as graphical models cannot capture the mutual exclusivity constraints associated with permutations in this paper we use the low frequency terms of a fourier decomposition to represent distributions over permutations compactly we present kronecker conditioning a novel approach for maintaining and updating these distributions directly in the fourier domain allowing for polynomial time bandlimited approximations low order fourier based approximations however may lead to functions that do not correspond to valid distributions to address this problem we present a quadratic program defined directly in the fourier domain for projecting the approximation onto a relaxation of the polytope of legal marginal distributions we demonstrate the effectiveness of our approach on a real camera based multi person tracking scenario

an open source lower limb model hip joint validation musculoskeletal lower limb models have been shown to be able to predict hip contact forces hcfs that are comparable to in vivo measurements obtained from instrumented prostheses however the muscle recruitment predicted by these models does not necessarily compare well to measured electromyographic emg signals in order to verify if it is possible to accurately estimate hcfs from muscle force patterns consistent with emg measurements a lower limb model based on a published anatomical dataset klein horsman et al 2007 clinical biomechanics 22 239 247 has been implemented in the open source software opensim a cycle to cycle hip joint validation was conducted against hcfs recorded during gait and stair climbing trials of four arthroplasty patients bergmann et al 2001 journal of biomechanics 34 859 871 hip joint muscle tensions were estimated by minimizing a polynomial function of the muscle forces the resulting muscle activation patterns obtained by assessing multiple powers of the objective function were compared against emg profiles from the literature calculated hcfs denoted a tendency to monotonically increase their magnitude when raising the power of the objective function the best estimation obtained from muscle forces consistent with experimental emg profiles was found when a quadratic objective function was minimized average overestimation at experimental peak frame 10 1 for walking 7 8 for stair climbing the lower limb model can produce appropriate balanced sets of muscle forces and joint contact forces that can be used in a range of applications requiring accurate quantification of both the developed model is available at the website https simtk org home low limb london

muscular synergism i on criteria for load sharing between synergistic muscles the use of optimization techniques to predict individual muscle forces in redundant biomechanical systems implies the formulation of a criterion for load sharing between the muscles in part i of this paper the characteristics and performance of several linear and non linear criteria reported in the literature have been compared for static isometric knee flexion the results show that linear criteria inherently predict discrete muscle action orderly recruitment of muscles whereas non linear criteria can predict synergistic action all criteria predict that relatively more force is allocated to muscles with large moment arms when muscle stresses or ratios of muscle force to maximum muscle force are used as the decision variables in the objective function then relatively more force is allocated to muscles with large maximum possible force as well future formulations of the optimization should consider the differences in fiber type composition among the muscles such an approach is presented in part ii of the paper

dynamic coordinate data for describing muscle tendon paths a mathematical approach when modelling the musculoskeletal system over a range of joint angles the use of fixed points to describe muscle tendon paths has inherent limitations these result in fewer deflection points and the use of effective insertions to accommodate both relative marker movement and avoid muscle paths contacting bony structures model performance is dependent on the joint angle relative to the anatomical position where the muscle tendon paths were defined the present study proposes a scheme for the implementation of dynamic coordinates for describing muscle tendon paths for each muscle tendon element a plane is defined in which the muscle tendon complex acts when crossing a given joint the muscle tendon plane is dependent on 3d segment orientations and describes one degree of freedom while the remaining two degrees of freedom are described by polar coordinates and locate the dynamic point in the muscle tendon plane the dynamic approach is implemented on four muscles of the lower limb in modelled and simulated joint movements and offers a significant improvement on previous approaches based on fixed deflection points the scheme accommodates compound 3d rotations about joint axes is not computationally difficult or require large data sets and does not impose limitations on the number of points that may be defined along a muscle tendon path

conformations of linear dna we examine the conformations of a model for underwound and overwound dna the molecule is represented as a cylindrically symmetric elastic string subjected to a stretching force and to constraints corresponding to a specification of the link number we derive a fundamental relation between the euler angles that describe the curve and the topological linking number analytical expressions for the spatial configurations of the molecule in the infinite length limit are obtained a unique configuration minimizes the energy for a given set of physical conditions an elastic model incorporating thermal fluctuations provides excellent agreement with experimental results on the plectonemic transition

writhe formulas and antipodal points in plectonemic dna configurations the linking and writhing numbers are key quantities when characterizing the structure of a piece of supercoiled dna defined as double integrals over the shape of the double helix these numbers are not always straightforward to compute though a simplified formula was established in a theorem by fuller proc natl acad sci u s a 75 3557 1978 we examine the range of applicability of this widely used simplified formula and show that it cannot be employed for plectonemic dna we show that inapplicability is due to a hypothesis of fuller theorem that is not met the hypothesis seems to have been overlooked in many works