smoothness of an equation for the glottal flow rate versus the glottal area this letter proposes a correction to an equation by titze j acoust soc am 75 570 580 1984 for the volume velocity of the glottal air flow given the glottal area and other laryngeal parameters it shows that the equation produces non differentiable waveforms at the instants of glottal closure and opening if the glottal area is also not differentiable at those instants by adding an air viscosity term to the equation twice differentiability is obtained also the letter corrects a sign error in the original formulation

direct numerical simulation of the glottal jet and vocal fold dynamics in a three dimensional laryngeal model an immersed boundary method based flow solver coupled with a finite element solid dynamics solver is employed in order to conduct direct numerical simulations of phonatory dynamics in a three dimensional model of the human larynx the computed features of the glottal flow including mean and peak flow rates and the open and skewness quotients are found to be within the normal physiological range the flow induced vibration pattern shows the classical convergent divergent glottal shape and the vibration amplitude is also found to be typical for human phonation the vocal fold motion is analyzed through the method of empirical eigenfunctions and this analysis indicates a 1 1 modal entrainment between the adduction abduction mode and the mucosal wave mode the glottal jet is found to exhibit noticeable cycle to cycle asymmetric deflections and the mechanism underlying this phenomenon is examined

bifurcations in an asymmetric vocal fold model a two mass model of vocal fold vibrations is analyzed with methods from nonlinear dynamics bifurcations are located in parameter planes of physiological interest subglottal pressure stiffness of the folds it is shown that a sufficiently large tension imbalance of the left and right vocal fold induces bifurcations to subharmonic regimes toroidal oscillations and chaos the corresponding attractors are characterized by phase portraits spectra and next maximum maps the relevance of these simulations for voice disorders such as laryngeal paralysis is discussed

chaotic component obscured by strong periodicity in voice production system the effect of glottal aerodynamics in producing the nonlinear characteristics of voice is investigated by comparing the outputs of the asymmetric composite model and the two mass model the two mass model assumes the glottal airflow to be laminar nonviscous and incompressible in this model when the asymmetric factor is decreased from 0 65 to 0 35 only 1 1 and 1 2 modes are detectable however with the same parameters four vibratory modes 1 1 1 2 2 4 2 6 are found in the asymmetric composite model using the navier stokes equations to describe the complex aerodynamics in the glottis moreover the amplitude of the waveform is modulated by a small amplitude noiselike series the nonlinear detection method reveals that this noiselike modulation is not random but rather it is deterministic chaos this result agrees with the phenomenon often seen in voice in which the voice signal is strongly periodic but modulated by a small amplitude chaotic component the only difference between the two mass model and the composite model is in their descriptions of glottal airflow therefore the complex aerodynamic characteristics of glottal airflow could be important in generating the nonlinear dynamic behavior of voice production including bifurcation and a small amplitude chaotic component obscured by strong periodicity

experimental validation of a three dimensional reduced order continuum model of phonation due to the complex nature of the phonation process a one dimensional 1d glottal flow description is often used in current phonation models although widely used in voice research these 1d flow based phonation models have not been rigorously validated against experiments in this study a 1d glottal flow model is coupled with a three dimensional nonlinear continuum model of the vocal fold and its predictions are compared to physical model experiments the results show that the 1d flow based model is able to predict the phonation threshold pressure and onset frequency within reasonable accuracy and to reproduce major vibratory features observed in the experiments

finite element approximation of flow induced vibrations of human vocal folds model effects of inflow boundary conditions and the length of subglottal and supraglottal channel on phonation onset abstract the paper presents the numerical analysis of interaction of the vibrating simplified human vocal folds model with the incompressible viscous airflow in a channel modeling simplified subglottal and supraglottal spaces the flow in the considered 2d computational fluid domain is governed by the navier stokes equations written in the arbitrary lagrangian eulerian form the stabilized finite element method is applied for numerical approximation and the choice of boundary conditions and their implementation is discussed for the considered model problem the prescribed inlet velocity and prescribed pressure difference formulations were numerically analyzed the prescribed inlet velocity formulation was successful in predicting of the flutter velocity value whereas the prescribed pressure difference gave nonphysical results finally a modified inlet boundary condition motivated by the penalization approach is suggested it is shown that this approach gives possibilities to optimize the inlet boundary condition related to a physical reality by changing smoothly the penalty parameter in the interval between the two extremes and to treat the complete closures of the channel

theoretical and experimental study of quasisteady flow separation within the glottis during phonation application to a modified two mass model most flow models used in numerical simulation of voiced sound production rely for the sake of simplicity upon a certain number of assumptions while most of these assumptions constitute reasonable first approximations others appear more doubtful in particular it is implicitly assumed that the air flow through the glottal channel separates from the walls at a fixed point since this assumption appears quite unrealistic and considering that the position of the separation point is an important parameter in phonation models in this paper a revised fluid mechanical description of the air flow through the glottis is proposed in which the separation point is allowed to move this theoretical model as well as the assumptions made are validated using steady and unsteady flow measurements combined with flow visualizations in order to evaluate the effective impact of the revised theory we then present an application to a simple mechanical model of the vocal cords derived from the classical two mass model as expected implementation of a moving separation point appears to be of great importance for the modeling of glottal signals it is further shown that the numerical model coupled with a more realistic description of the vocal cord collision can lead to signals surprisingly close to those observed in real speech by inverse filtering

the birational type of the moduli space of even spin curves abstract we determine the kodaira dimension of the moduli space s g of even spin curves for all g precisely we show that s g is of general type for g 8 and has negative kodaira dimension for g 8

determination of parameters for lumped parameter models of the vocal folds using a finite element method approach to study the mechanical behavior of the vocal folds lumped parameter models of the vocal folds have been developed in the past coupling with a model of the aerodynamics in the glottis provides the possibility of simulating glottal waves a new method is presented to obtain values for the masses and springs of the lumped parameter models by using a finite element method model of the vocal folds this finite element method model is based on geometry and material data from the literature resulting in a model that describes the vocal fold in a realistic way requiring the dynamic behavior of the lumped parameter model of the vocal fold to be equal to the dynamic behavior of the finite element method model of the vocal fold parameter values are obtained that are purely based on dynamic considerations therefore the behavior of the vocal folds is described in a realistic way by these parameters these values are compared with the values used by previous authors and are implemented in lumped parameter models self sustained oscillation is achieved with the new values for masses and springs

trace class operators and commutators abstract we identify the linear span of commutators ab ba where a is a trace class operator and b is any bounded operator on a separable hilbert space or where a b are both hilbert schmidt operators thus answering questions raised by work of anderson vaserstein and weiss

spin curves and scorza quartics in the paper takagi and zucconi in on blow ups of the quintic del pezzo 3 fold and varieties of power sums of quartic hypersurfaces pp 1 40 preprint submitted 2008 we construct new subvarieties in the varieties of power sums for certain quartic hypersurfaces in this paper we show that these quartics coincide with the scorza quartics of general pairs of trigonal curves and ineffective theta characteristics among other applications we give an affirmative answer to the conjecture of dolgachev and kanev on the existence of the scorza quartics for any general pairs of curves and ineffective theta characteristics we also give descriptions of the moduli spaces of trigonal even spin curves

a theoretical study of the hysteresis phenomenon at vocal fold oscillation onset offset this paper presents a theoretical study on the differences in the biomechanical parameters of the vocal folds between oscillation onset and offset the dynamics of the oscillation is analyzed from the perspective of the theory of nonlinear dynamical systems using a mucosal wave model of the vocal folds with the subglottal pressure and the vocal fold half width as control parameters it is shown that the oscillation onset occurs through a hopf bifurcation of the subcritical type at which an unstable limit cycle is generated also the oscillation offset occurs at a cyclic fold bifurcation at which the unstable limit cycle and a stable limit cycle the actual vocal fold oscillation coalesce and cancel each other both bifurcations combine to form an oscillation hysteresis phenomenon common in cases of flow induced oscillations an analytical expression for the onset offset ratio of parameters is derived the onset offset ratio is in the range of 0 5 1 in agreement with the experimental evidence this value depends on the phase delay in motion of the upper edge of the vocal folds versus the lower edge and on the particular model adopted for airflow separation within the glottis

which traces are spectral abstract among ideals of compact operators on a hilbert space we identify a subclass of those closed with respect to the logarithmic submajorization within this subclass we answer the questions asked by pietsch 22 and by dykema figiel weiss and wodzicki 7 in the first case we show that lidskii type formulae hold for every trace on such ideal in the second case we provide the description of the commutator subspace associated with a given ideal finally we prove that a positive trace on an arbitrary ideal is spectral if and only if it is monotone with respect to the logarithmic submajorization

the moduli space of genus four even spin curves is rational abstract using the mori theory for threefolds we prove that the moduli space of pairs of smooth curves of genus four and theta characteristics without global sections is a rational variety

the action functional in non commutative geometry we establish the equality between the restriction of the adler manin wodzicki residue or non commutative residue to pseudodifferential operators of order n on ann dimensional compact manifoldm with the trace which j dixmier constructed on the macaev ideal we then use the latter trace to recover the yang mills interaction in the context of non commutative differential geometry

an in vitro setup to test the relevance and the accuracy of low order vocal folds models an experimental setup and human vocal folds replica able to produce self sustained oscillations are presented the aim of the setup is to assess the relevance and the accuracy of theoretical vocal folds models the applied reduced mechanical models are a variation of the classical two mass model and a simplification inspired on the delayed mass model for which the coupling between the masses is expressed as a fixed time delay the airflow is described as a laminar flow with flow separation the influence of a downstream resonator is taken into account the oscillation pressure threshold and fundamental frequency are predicted by applying a stability analysis to the mechanical models the measured frequency response of the mechanical replica together with the initial rest area allows us to determine the model parameters spring stiffness damping geometry masses validation of theoretical model predictions to experimental data shows the relevance of low order models in gaining a qualitative understanding of phonation however quantitative discrepancies remain large due to an inaccurate estimation of the model parameters and the crudeness in either flow or mechanical model description as an illustration it is shown that significant improvements can be made by accounting for viscous flow effects

modeling of chaotic vibrations in symmetric vocal folds the chaos mechanism of above range phonation was examined in symmetrically modeled vocal folds by using the traditional two mass model assumption the poincare map technique was used to display chaotic attractors this method provided an effective description of irregular vocal fold movements the power spectrum lyapunov exponent and kaplan yorke dimension were employed to describe chaotic vibrations in the vocal fold model these nonlinear dynamic analyses suggested that for the positive lyapunov exponent chaotic attractors contribute to irregular vocal fold vibrations descriptions of complicated irregular vibrations of the vocal fold yielded evidence of chaos to investigate the effects of independent parameters such as subglottal pressure coupling stiffness and phonation neutral area bifurcation diagrams based on the poincare map were discussed the results confirmed that the dynamics of the two mass model was strongly influenced by independent parameters nonlinear dynamic methods were expecte

the effect of glottal angle on intraglottal pressure intraglottal pressure distributions depend upon glottal shape size and diameter this study reports the effects of varying glottal angle on intraglottal and transglottal pressures using a three dimensional plexiglas model with a glottis having nine symmetric glottal angles and a constant minimal glottal diameter of 0 06cm the empirical data were supported by computational results using fluent the results suggested that 1 the greater the convergent glottal angle the greater outward driving forces higher intraglottal pressures on the vocal folds 2 flow resistance was greatest for the uniform glottis and least for the 10 divergent glottis 3 the greatest negative pressure in the glottis and therefore the greatest pressure recovery for diverging glottal shapes occurred for an angle of 10 4 the smaller the convergent angle the greater the flow resistance 5 fluent was highly accurate in predicting the empirical pressures of this model 6 flow separation locations given by fluent for

traces of compact operators and the noncommutative residue abstract we extend the noncommutative residue of m wodzicki on compactly supported classical pseudo differential operators of order d and generalise a connes trace theorem which states that the residue can be calculated using a singular trace on compact operators contrary to the role of the noncommutative residue for the classical pseudo differential operators a corollary is that the pseudo differential operators of order d do not have a unique trace pseudo differential operators can be non measurable in connes sense other corollaries are given clarifying the role of dixmier traces in noncommutative geometry including the definitive statement of connes original theorem

numerical simulation of self sustained oscillation of a voice producing element based on navier stokes equations and the finite element method surgical removal of the larynx results in radically reduced production of voice and speech to improve voice quality a voice producing element vpe is developed based on the lip principle called after the lips of a musician while playing a brass instrument to optimize the vpe a numerical model is developed in this model the finite element method is used to describe the mechanical behavior of the vpe the flow is described by two dimensional incompressible navier stokes equations the interaction between vpe and airflow is modeled by placing the grid of the vpe model in the grid of the aerodynamical model and requiring continuity of forces and velocities by applying and increasing pressure to the numerical model pulses comparable to glottal volume velocity waveforms are obtained by variation of geometric parameters their influence can be determined to validate this numerical model an in vitro test with a prototype of the vpe is performed experimental and numerical results show an acceptable agreement

experimental validation of a quasi steady theory for the flow through the glottis in this paper a theoretical description of the flow through the glottis based on a quasi steady boundary layer theory is presented the thwaites method is used to solve the von karman equations within the boundary layers in practice this makes the theory much easier to use compared to pohlhausen s polynomial approximations this theoretical description is evaluated on the basis of systematic comparison with experimental data obtained under steady flow or unsteady oscillating flow without and with moving vocal folds results tend to show that the theory reasonably explains the measured data except when unsteady or viscous terms become predominant this happens particularly during the collision of the vocal folds

numerical simulations of fluid structure interactions in single reed mouthpieces most single reed woodwind instrument models rely on a quasistationary approximation to describe the relationship between the volume flow and the pressure difference across the reed channel semiempirical models based on the quasistationary approximation are very useful in explaining the fundamental characteristics of this family of instruments such as self sustained oscillations and threshold of blowing pressure however they fail at explaining more complex phenomena associated with the fluid structure interaction during dynamic flow regimes such as the transient and steady state behavior of the system as a function of the mouthpiece geometry previous studies have discussed the accuracy of the quasistationary approximation but the amount of literature on the subject is sparse mainly due to the difficulties involved in the measurement of dynamic flows in channels with an oscillating reed in this paper a numerical technique based on the lattice boltzmann method and a finite difference scheme is proposed in order to investigate the characteristics of fully coupled fluid structure interaction in single reed mouthpieces with different channel configurations results obtained for a stationary simulation with a static reed agree very well with those predicted by the literature based on the quasistationary approximation however simulations carried out for a dynamic regime with an oscillating reed show that the phenomenon associated with flow detachment and reattachment diverges considerably from the theoretical assumptions furthermore in the case of long reed channels the results obtained for the vena contracta factor are in significant disagreement with those predicted by theory for short channels the assumption of constant vena contracta was found to be valid for only 40 of the duty cycle

source tract interaction with prescribed vocal fold motion an equation describing the time evolution of glottal volume velocity with specified vocal fold motion is derived when the sub and supra glottal vocal tracts are present the derivation of this fant equation employs a property explicated in howe and mcgowan 2011 j fluid mech 672 428 450 that the fant equation is the adjoint to the equation characterizing the matching conditions of sub and supra glottal green s functions segments with the glottal segment the present aeroacoustic development shows that measurable quantities such as input impedances at the glottis provide the coefficients for the fant equation when source tract interaction is included in the development explicit expressions for the green s function are not required with the poles and zeros of the input impedance functions specified the fant equation can be solved after the general derivation of the fant equation the specific cases where plane wave acoustic propagation is described either by a sturm liouville problem or concatenated cylindrical tubes is considered simulations show the expected skewing of the glottal volume velocity pulses depending on whether the fundamental frequency is below or above a sub or supra glottal formant more complex glottal wave forms result when both the first supra glottal fundamental frequencies are high and close to the first sub glottal formant

glottal flow through a two mass model comparison of navier stokes solutions with simplified models a new numerical model of the vocal folds is presented based on the well known two mass models of the vocal folds the two mass model is coupled to a model of glottal airflow based on the incompressible navier stokes equations glottal waves are produced using different initial glottal gaps and different subglottal pressures fundamental frequency glottal peak flow and closed phase of the glottal waves have been compared with values known from the literature the phonation threshold pressure was determined for different initial glottal gaps the phonation threshold pressure obtained using the flow model with navier stokes equations corresponds better to values determined in normal phonation than the phonation threshold pressure obtained using the flow model based on the bernoulli equation using the navier stokes equations an increase of the subglottal pressure causes the fundamental frequency and the glottal peak flow to increase whereas the fundamental frequency in the bernoulli based model does not change with increasing pressure

increasing the complexity of glottal flow models in vitro validation for steady flow conditions quasi one dimensional glottal flow descriptions predict vocal folds oscillations characteristics which are qualitatively relevant to in vitro and in vivo experimental data the current paper considers the resolution of the 2d navier stokes equations in order to obtain a refined description of the flow phenomena adapted to more realistic glottal geometry the pressure and flow rate predictions obtained from quasi one dimensional flow models and the resolution of the 2d navier stokes equations are examined for steady flows within a rigid glottis the models predictions are validated against in vitro measurements performed on rigid constriction replicas comparable to the geometrical conditions of the glottis and mounted in a suitable set up the confrontation between the experimental and computed data tends to show that the accuracy of the estimated pressures increases with the complexity of the flow model whereas the inverse tendency can be observed for the estimated flow rates a focus is made on the flow

low dimensional models of the glottal flow incorporating viscous inviscid interaction the behavior of glottal flow can to a large extent be characterized by development and separation of the boundary layer the point of flow separation is known to vary during the phonatory cycle due to change in channel configuration to take the movable nature of the separation point into account the boundary layer equation is solved numerically and the values of the characteristic quantities are determined as well as the point of separation development of the boundary layer in general reduces the effective size of the channel and therefore increases the core flow velocity which in turn provides the boundary condition of the boundary layer equation the interaction between the viscous boundary layer and inviscid core flow parts of the glottal flow is therefore strongly indicated to apply this viscous inviscid interaction the expression of the core flow is derived for a two dimensional flow field and is solved jointly with the boundary layer equation numerical results are shown to examine the effect of the reynolds number and glottal configuration with special emphasis on the comparison of flow models developed for one and two dimensional flow fields numerical results are also quantitatively compared with data obtained from flow measurement experiments

aeroacoustic source characterization in a physical model of phonation this paper presents measurements conducted in a physical model of the adult human airway the goals of this work are to 1 benchmark the physical model to excised larynx models in the literature and 2 empirically demonstrate the relationship between vocal fold drag and sound production results from the airway model are first benchmarked to published time averaged behavior of excised larynx models the airway model in this work exhibited higher glottal volume flow lower glottal resistance and less fundamental frequency variation than excised larynx models next concurrent measurements of source behavior and radiated sound were compared unsteady transglottal pressure a surrogate measure for vocal fold drag and radiated sound measured at the mouth showed good correlation in particular the standard deviation and the ratio of the power of the first and second harmonics of the transglottal and mouth pressures were strongly correlated this empirical result supports the assertion that vocal fold drag is the principal source of sound in phonation

intraglottal pressure profiles for a symmetric and oblique glottis with a divergence angle of 10 degrees human phonation does not always involve symmetric motions of the two vocal folds asymmetric motions can create slanted or oblique glottal angles this study reports intraglottal pressure profiles for a plexiglas model of the larynx with a glottis having a 10 degree divergence angle and either a symmetric orientation or an oblique angle of 15 degrees for the oblique glottis one side was divergent and the other convergent the vocal fold surfaces had 14 pressure taps the minimal glottal diameter was held constant at 0 04 cm results indicated that for either the symmetric or oblique case the pressure profiles were different on the two sides of the glottis except for the symmetric geometry for a transglottal pressure of 3 cm h2o for the symmetric case flow separation created lower pressures on the side where the flow stayed attached to the wall and the largest pressure differences between the two sides of the channel were 5 6 of the transglottal pressure for the oblique case pressures were lower

a complex ball uniformization of the moduli space of cubic surfaces via periods of k3 surfaces in this paper we show that the moduli space of nodal cubic surfaces is iso morphic to a quotient of a 4 dimensional complex ball by an arithmetic subgroup of the unitary group this complex ball uniformization uses the periods of certain k3 surfaces which are naturally associated to cubic surfaces a similar uniformization is given for dif ferent covers of the moduli space corresponding to geometric markings of the picard group or a choice of a line on the surface we also give a detailed description of the boundary com ponents corresponding to singular surfaces

influence of collision on the flow through in vitro rigid models of the vocal folds measurements of pressure in oscillating rigid replicas of vocal folds are presented the pressure upstream of the replica is used as input to various theoretical approximations to predict the pressure within the glottis as the vocal folds collide the classical quasisteady boundary layer theory fails it appears however that for physiologically reasonable shapes of the replicas viscous effects are more important than the influence of the flow unsteadiness due to the wall movement a simple model based on a quasisteady bernoulli equation corrected for viscous effect combined with a simple boundary layer separation model does globally predict the observed pressure behavior

viscous effects in a static physical model of the uniform glottis the classic work on laryngeal flow resistance by van den berg et al j acoust soc am 29 626 631 1957 is revisited these authors used a formula to summarize their measurements and thus they separated the effects of entrance loss and pressure recovery from those of viscosity within the glottis analysis of intraglottal pressure distributions obtained from the physical model m5 r scherer et al j acoust soc am 109 1616 1630 2001 reveals substantial regions within the glottis where the pressure gradient is almost constant for glottal diameters from 0 005 to 0 16 cm as expected when viscous effects dominate the flow resistance of a narrow channel for this set of glottal diameters the part of the pressure gradient that has a linear dependence on the glottal volume velocity is isolated the inverse cube diameter of the poiseuille expression for glottal flows is examined with the data set provided by the m5 intraglottal pressure distributions the poiseuille effect is found to give a reasonable account of viscous effects in the diameter interval from 0 0075 to 0 02 cm but an inverse 2 59 power law gives a closer fit across all diameters

influence of glottal cross section shape on phonation onset phonation models commonly rely on the assumption of a two dimensional glottal geometry to assess kinetic and viscous flow losses in this paper the glottal cross section shape is taken into account in the flow model in order to capture its influence on vocal folds oscillation for the assessed cross section shapes rectangular elliptical or circular segment the minimum pressure threshold enabling to sustain vocal folds oscillation is altered for constriction degrees smaller than 75 the discrepancy between cross section shapes increases as the constriction degree decreases

sums of commutators in non commutative banach function spaces let m be a ii factor and denote by its normal faithful semi finite trace for any rearrangement invariant kothe function space x on 0 let x m be the associated non commutative banach function space this paper is concerned with ideals in m of the form ix m m x m that are contained in lp m for some p 0 it is proved that an element t in ix m is a finite sum of commutators of the form a b with a ix m and b m if and only if the function t 1t t t d t belongs to x where t is the brown spectral measure of t and t t t is the non increasing rearrangement of the function with respect to t this extends to general banach function spaces a result obtained by kalton for quasi banach ideals of compact operators and implies that the dixmier s trace of a quasi nilpotent element in l1 m is always zero

phonation threshold pressure a missing link in glottal aerodynamics phonation threshold pressure has previously been defined as the minimum lung pressure required to initiate phonation by modeling the dependence of this pressure on fundamental frequency it is shown that relatively simple aerodynamic relations for time varying flow in the glottis are obtained lung pressure and peak glottal flow are nearly linearly related but not proportional for this reason typical power law relations that have previously been proposed do not hold glottal impedance for time varying flow must be defined differentially rather than as a simple ratio between pressure and flow it is shown that the peak flow the peak flow derivative the open quotient and the speed quotient of inverse filtered glottal flow waveforms all depend explicitly on phonation threshold pressure data from singers are compared with those from nonsingers the primary difference is that singers obtain two to three times greater peak flow for a given lung pressure suggesting that they adjust their glottal or voca

commutator structure of operator ideals commutator structure of operator ideals ken dykema 1 tadeusz figiel gary weiss and mariusz wodzicki department of mathematics texas a m university college station tx 77843 3368 usa b instytut matematyczny polskiej akademii nauk 81 825 sopot poland department of mathematical sciences university of cincinnati cincinnati oh 45221 0025 usa department of mathematics university of california berkeley ca 94720 3840 usa

on the generalised fant equation abstract an analysis is made of the fluid structure interactions involved in the production of voiced speech it is usual to avoid time consuming numerical simulations of the aeroacoustics of the vocal tract and glottis by the introduction of fant s reduced complexity equation for the glottis volume velocity q g fant acoustic theory of speech production mouton the hague 1960 a systematic derivation is given of fant s equation based on the nominally exact equations of aerodynamic sound this can be done with a degree of approximation that depends only on the accuracy with which the time varying flow geometry and surface acoustic boundary conditions can be specified and replaces fant s original lumped element heuristic approach the method determines all of the effective source terms governing q it is illustrated by consideration of a simplified model of the vocal system involving a self sustaining single mass model of the vocal folds that uses free streamline theory to account for surface friction and flow separation within the glottis identification is made of a new source term associated with the unsteady vocal fold drag produced by their oscillatory motion transverse to the mean flow

non commutative banach function spaces in this paper we survey some aspects of the theory of non commutative banach function spaces that is spaces of measurable operators associated with a semi finite von neumann algebra these spaces are also known as non commutative symmetric spaces the theory of such spaces emerged as a common generalization of the theory of classical commutative rearrangement invariant banach function spaces in the sense of w a j luxemburg and a c zaanen and of the theory of symmetrically normed ideals of bounded linear operators in hilbert space in the sense of i c gohberg and m g krein these two cases may be considered as the two extremes of the theory in the first case the underlying von neumann algebra is the commutative algebra l on some measure space with integration as trace in the second case the underlying von neumann algebra is b the algebra of all bounded linear operators on a hilbert space with standard trace important special cases of these non commutative spaces are the non commutative l p spaces which correspond in the commutative case with the usual l p spaces on a measure space and in the setting of symmetrically normed operator ideals they correspond to the schatten p classes mathfrak s p