the generalized roper suffridge extension operator in banach spaces ii abstract in this paper we generalize the roper suffridge extension operator from c n to banach spaces it is proved that this operator preserves the biholomorphic starlikeness on some domains in banach spaces from these we may construct a lots of concrete examples about biholomorphic starlike mappings on some domains in c n or hilbert spaces or banach spaces from univalent starlike functions on the unit disc u in c meanwhile the growth theorems of the corresponding mappings are given some results of gong and liu roper and suffridge graham et al in c n are extended to hilbert spaces or banach spaces

loewner chains and nonlinear resolvents of the caratheodory family on the unit ball in cn abstract in this paper we study various properties of nonlinear resolvents of holomorphic mappings in the caratheodory family m b n where b n is the euclidean unit ball in c n first we prove certain characterizations of inverse loewner chains f z t e 0 t a d z on b n 0 where a 0 c is a locally lebesgue integrable function such that a t 0 for a e t 0 and which satisfies a natural assumption next we prove that if f m b n then the nonlinear resolvent family j r r 0 is an inverse loewner chain on b n and the associated herglotz vector field is of divergence type where j r j r f i n r f 1 r 0 this result is a generalization to higher dimensions of a recent result due to elin shoikhet and sugawa we prove that 1 r j r can be embedded as the first element of a normal loewner chain and the family 1 r j r f 0 r f m b n is a compact subset of s 0 b n also we deduce that the shearing of 1 r j r r 0 associated with the family m b 2 is quasi convex of type a and also starlike of order 4 5 we give a sufficient condition for 1 r j r to be quasiconformal on b n and to be extended to a quasiconformal homeomorphism of c n onto itself finally sharp coefficient bounds for the nonlinear resolvent families of certain subsets of m b n and also examples of support points of the compact families generated by nonlinear resolvent mappings on b 2 will be obtained

gaussian quadrature involving einstein and fermi functions with an application to summation of series polynomials 7rk rk d x k 0 1 2 are constructed which are orthogonal with respect to the weight distributions da t t et 1 dt and da t 1 et 1 dt r 1 2 on 0 oo moment related methods being inadequate a discretized stieltjes procedure is used to generate the coefficients ak pk in the recursion formula 7rk 1 t t ak k t fk7rk1 t k 0 1 2 so t 1 17j l t 0 the discretization is effected by the gauss laguerre and a composite fejer quadrature rule respectively numerical values of ak ipk as well as associated error constants are provided for 0 k 39 these allow the construction of gaussian quadrature formulae including error terms with up to 40 points examples of n point formulae n 5 5 40 are provided in the supplements section at the end of this issue such quadrature formulae may prove useful in solid state physics calculations and can also be applied to sum slowly convergent series

the frequency scale of speech intonation in intonation research prominence lending pitch movements have either been described on a linear or on a logarithmic frequency scale an experiment has been carried out to check whether pitch movements in speech intonation are perceived on one of these two scales or on a psychoacoustic scale representing the frequency selectivity of the auditory system this last scale is intermediary between the other two scales subjects matched the excursion size of prominence lending pitch movements in utterances resynthesized in different pitch registers their task was to adjust the excursion size in a comparison stimulus in such a way that it lent equal prominence to the corresponding syllable in a fixed test stimulus the comparison stimulus and the test stimulus had pitches running parallel on either the logarithmic frequency scale the psychoacoustic scale or the linear frequency scale in one half of the experimental sessions the test stimulus was presented in the low register while the comparison stimulus was presented in the high register and conversely for the other half of the sessions the result is that in all cases stimuli are matched in such a way that the average excursion sizes in different registers are equal on the psychoacoustic scale

on the precise growth covering and distortion theorems for normalized biholomorphic mappings abstract in this paper we consider a normalized biholomorphic mapping f x defined on the unit ball in a complex banach space where the origin 0 is a zero of order k 1 of f x x the precise growth and covering theorem for f x is obtained when f x is a starlike mapping or a starlike mapping of order especially the precise growth and covering theorem for f x is also established when f x is a quasi convex mapping moreover the precise distortion theorem for f x is given when f x is a convex mapping our result includes many known results

criteria for biholomorphic convex mappings on the unit ball in hilbert spaces in this paper we give a necessary and sufficient condition that a locally biholomorphic mapping f on the unit ball b in a complex hilbert space x is a biholomorphic convex mapping which improves some results of hamada and kohr and solves the problem which is posed by graham and kohr from this we derive some sufficient conditions for biholomorphic convex mapping we also introduce a linear operator in purpose to construct some concrete examples of biholomorphic convex mappings on b in hilbert spaces moreover we give some examples of biholomorphic convex mappings on b in hilbert spaces

evaluating human neural tuning curves from a mechanical model of the cochlea by relating them to psychophysical masking data human neural threshold tuning curves are estimated by scaling the parameters of allen s j acoust soc am 68 1660 1670 1980 resonant tectorial membrane model of the cat cochlea a way to evaluate the derived tuning curves using psychophysical data is developed based on a psychophysical detection model which relates the physiological tuning curves to psychophysical masking data a detection criterion defined by a relationship among the bandwidth of the frequency tuning curves expressed as an equivalent rectangular bandwidth erb the width of the excitation patterns expressed as an equivalent rectangular spread ers and the psychophysical critical ratio is explored and verified using cat data the detection criterion is then used to test the derived human curves by making predictions of psychophysical masking and comparing the predictions to experimental data the detection model may also provide a deeper understanding of the frequency resolving properties of the cochlea

locus of generation for the 2 f1 f2 vs 2 f2 f1 distortion product otoacoustic emissions in normal hearing humans revealed by suppression tuning onset latencies and amplitude correlations the present study used distortion product otoacoustic emission dpoae suppression tuning curves stcs dpoae onset latencies ols and dpoae amplitude correlations to investigate the locus of generation of the 2 f1 f2 dpoae versus the 2 f2 f1 dpoae in humans the results of the tuning study revealed that for the 2 f1 f2 dpoae the tips of the stcs tuned consistently below the geometric mean gm frequency of the primary tones in contrast for the 2 f2 f1 dpoae stcs tuned above the gm of the primaries with 50 of the tip frequencies at or above the 2 f2 f1 frequency place when the average ratio of the 2 f2 f1 to the 2 f1 f2 tip frequencies was computed a factor of 1 44 provided an estimate of the frequency shift needed to align the two dpoae generation sites other results showed that ols for the 2 f2 f1 dpoae were uniformly shorter than those for the 2 f1 f2 with differences at the low frequencies amounting to as much as 6 7 ms further for both dpoaes curves describing latency decreases as

on e quasi convex mappings in the unit ball of a complex banach space abstract in this paper a new class of biholomorphic mappings named e quasi convex mapping is introduced in the unit ball of a complex banach space meanwhile the definition of e starlike mapping is generalized from e 0 1 to e 1 1 it is proved that the class of e quasi convex mappings is a proper subset of the class of starlike mappings and contains the class of e starlike mappings properly for some e 1 0 0 1 we give a geometric explanation for e starlike mapping with e 1 1 and prove that the generalized roper suffridge extension operator preserves the biholomorphic e starlikeness on some domains in banach spaces for e 1 1 we also give some concrete examples of e quasi convex mappings or e starlike mappings for e 1 1 in banach spaces or c n furthermore some other properties of e quasi convex mapping or e starlike mapping are obtained these results generalize the related works of some authors

some sufficient conditions for biholomorphic convex mappings on bnp abstract a few biholomorphic convex mappings on the unit ball in c n for n 1 are known in this paper we shall prove some sufficient conditions for biholomorphic convex mappings on reinhardt domain b p n in c n some results of roper and suffridge hamada and kohr on b 2 n are extended to b p n from these we may construct some concrete biholomorphic convex mappings on b p n

loewner chains and parametric representation in several complex variables abstract let b be the unit ball of c n with respect to an arbitrary norm we study certain properties of loewner chains and their transition mappings on the unit ball b we show that any loewner chain f z t and the transition mapping v z s t associated to f z t satisfy locally lipschitz conditions in t locally uniformly with respect to z b moreover we prove that a mapping f h b has parametric representation if and only if there exists a loewner chain f z t such that the family e tf z t t 0 is a normal family on b and f z f z 0 for z b also we show that univalent solutions f z t of the generalized loewner differential equation in higher dimensions are unique when e tf z t t 0 is a normal family on b finally we show that the set s0 b of mappings which have parametric representation on b is compact

a class of slowly convergent series and their summation by gaussian quadrature series are considered whose general term is a rational function multiplied by a fractional power the summation of such series is reduced via laplace transformation techniques to a problem of quadrature which is then solved by gaussian quadrature relative to einstein and fermi weight functions a number of examples are worked out in detail

invariance principles for cochlear mechanics hearing phases a functional model of the cochlea is devised on the basis of the results from classical experiments the basilar membrane filter is investigated in detail its phase is close to linear in the region around the peak of the amplification on one side this has consequences for the time analysis and on the other side this has led to a prediction on phase perception for very simple combinations of tones a prediction which is now confirmed by experiments equivariance under the dilation group permits one to describe the model by a wavelet transform daubechies ten lectures on wavelets siam philadelphia 1992 the wavelet is discussed in reference to the phase analysis of the basilar membrane filter

the generalized roper suffridge extension operator abstract let f z be a normalized convex starlike function on the unit disc d let z c n z 1 2 z 2 p 2 z n p n where z z1 z2 zn z1 d z 2 z n c n 1 pi 1 i 2 n are real numbers in this note we prove that f z f z1 f z1 1 p2z2 f z1 1 pnzn is a normalized convex starlike mapping on where we choose the power function such that f z1 1 pi z1 0 1 i 2 n some other related results are proved

biholomorphic convex mappings of order α on the unit ball in hilbert spaces abstract in this paper we introduce the concept of biholomorphic convex mapping of order on the unit ball b in a complex hilbert space x then we provide some sufficient conditions that a locally biholomorphic mapping f is a biholomorphic convex mapping of order we give an alexander type theorem between the subclass of convex mappings and the subclass of starlike mappings on b in a hilbert space we also study the order of starlikeness of biholomorphic convex mappings of order on b in a hilbert space finally we construct some examples of biholomorphic convex mappings of order on b in a hilbert space by a linear operator

convex subordination chains and injective mappings in cn abstract in this paper we continue the work related to convex subordination chains in c and c n and prove that if f z z k 2 a k z k is a holomorphic mapping on the euclidean unit ball b n in c n such that k 2 k 2 a k 1 a 0 1 0 is a function of class c 2 on 0 1 and continuous on 0 1 such that a 1 0 a t 0 t a t 1 2 for t 0 1 and if a satisfies a differential equation on 0 1 then f z t a t 2 d f t z t z f t z is a convex subordination chain over 0 1 and the mapping f z a z 2 d f z z f z is injective on b n we also present certain coefficient bounds which provide sufficient conditions for univalence quasiregularity and starlikeness for the chain f z t finally we give some examples of convex subordination chains over 0 1

the spiral of theodorus numerical analysis and special functions theodorus of cyrene ca 460 399 b c teacher of plato und theaetetus is known for his proof of the irrationality of n n 2 3 5 17 he may have known also of a discrete spiral today named after him whose construction is based on the square roots of the numbers n 1 2 3 the subject of this lecture is the problem of interpolating this discrete angular spiral by a smooth if possible analytic spiral an interesting solution was proposed in 1993 by p j davis which is based on an infinite product the computation of this product gives rise to problems of numerical analysis in particular the summation of slowly convergent series and the identification of the product raises questions regarding special functions the former are solved by a method of integration in particular gaussian integration the latter by means of dawson s integral und the bose einstein distribution number theoretic questions also loom behind this work

the cochlear frequency map for the cat labeling auditory nerve fibers of known characteristic frequency iontophoresis of horseradish peroxidase was used to label single auditory nerve fibers after determination of threshold tuning curves and rates of spontaneous discharge the relation between characteristic frequency cf and cochlear longitudinal location is reconstructed from 52 labeled neurons in 16 cochleas the length of the organ of corti alotted to an octave of stimulus frequency increases steadily from low to high frequencies thus there is not a simple linear distance to log frequency conversion when comparing cochleas of different total length the best predictor of cf at a given location is the distance from base or apex expressed as a percentage of the total length the cochlear frequency map derived from these single neuron labeling experiments is compared to maps derived by a number of different physiological and psychophysical techniques and the significance of the similarities and differences is discussed

the mammalian cochlear map is optimally warped the form of the mammalian cochlear frequency position map has been well described by greenwood and empirical values found for its coefficients for a number of species the apical portion of the mammalian map is spatially compressed relative to the base and this nonuniformity in the representation of frequency is evidently consistent across species however an evolutionary reason for this consistency encompassing critical band behavior with respect to position is conspicuously missing likewise the length of the cochlea in any mammal including echolocating species is related to body size but attempts to explain the length in terms of frequency limits range or resolution have no general explanation new insight stems from a hypothesis in which the map curvature may be appreciated as an adaptation for optimal frequency resolution over the auditory range it is demonstrated numerically that the mammalian curve may be considered a member of a family of curves which vary in their degree of warp the warp factor found to be common across mammals is an optimal trade off between four conflicting constraints 1 enhancing high frequency resolution 2 setting a lower bound on loss of existing low frequency resolution 3 minimizing map nonuniformity and 4 keeping the whole map smooth thereby avoiding reflections

frequency representation in the emu basilar papilla the frequency representation on the basilar papilla of the emu was investigated by recording from single auditory afferents and labeling them iontophoretically with horseradish peroxidase successfully labeled fibers covered a range of characteristic frequencies from 0 05 to 3 3 khz which corresponded to nearly 90 of the total papillar length the termination sites of labeled fibers within the basilar papilla correlated with their characteristic frequency the lowest frequencies being represented apically the highest basally the best mathematical description of the frequency distribution was given by an exponential regression with a uniform mapping constant of 0 8 mm octave this function deviates from the more commonly observed pattern of increasing space constant towards higher frequencies most labeled fibers innervated only one hair cell those contacting 4 or more hair cells were all of low characteristic frequency 0 11 khz all labeled fibers terminated within the neural 56 of the basilar papilla s width this innervation pattern is consistent with that observed in other bird species there was no conclusive evidence for a change of sensitivity and or frequency selectivity with innervational position across the papilla s width

duration sensation in relation to the autocorrelation function of sound stimuli relationships were investigated between subjective judgment of physical duration and the factors extracted from the autocorrelation function acf of sound stimuli factors of acf were changed by using different properties of pure tone complex tone white noise and bandpass noise stimuli experiments by paired comparison method were conducted in the sound proof chamber except for one in a concert hall sound stimuli were selected on the basis of the measured factors of acf as parameters subjects were asked to compare the durations of two stimuli in the pair subjective durations were obtained in the psychometric function and were interpreted as duration sensation ds on the basis of the auditory brain model and the theory of primary sensation results p 0 05 are shown as below 1 dss are longer with larger 1 being time delay of first peak of acf corresponding to pitch 2 dss are longer with higher 0 being energy at the beginning of acf 3 dss are longer with smaller m being constant am

applications of a decomposition of holomorphic mappings in cn with respect to a cyclic group abstract in the paper the decomposition with respect to the group of roots of unity is applied to characterize some classes of biholomorphic mappings in c n there is considered the problem of subordination and majorization relations between convex mappings and their components in the above partition there is also given a distortion theorem for convex mappings and a criterion for starlikeness in c n

on a subclass of quasi convex mappings abstract a class of biholomorphic mappings named quasi convex mappings of type d is introduced in the unit ball of a complex banach space which is the natural generalization of the convex functions in one variable it is proved that this class of mappings is a subset of the class of quasi convex mappings of type a introduced by k a roper and t j suffridge

the sharp estimates of homogeneous expansions for the generalized class of close to quasi convex mappings abstract in this paper the class of strongly close to quasi convex mappings of type and order is introduced in the unit ball of complex banach space or unit polydisk in c n and we obtain the sharp estimates of homogeneous expansions for this class these results generalize many known results

a phenomenological model for the responses of auditory nerve fibers i nonlinear tuning with compression and suppression a phenomenological model was developed to describe responses of high spontaneous rate auditory nerve an fibers including several nonlinear response properties level dependent gain compression bandwidth and phase properties were implemented with a control path that varied the gain and bandwidth of tuning in the signal path filter by making the bandwidth of the control path broad with respect to the signal path the wide frequency range of two tone suppression was included by making the control path filter level dependent and tuned to a frequency slightly higher than the signal path filter other properties of two tone suppression were also included these properties included the asymmetrical growth of suppression above and below the characteristic frequency and the frequency offset of the suppression tuning curve with respect to the excitatory tuning curve the implementation of this model represents a relatively simple phenomenological description of a single mechanism that underlies several important nonlinear response properties of an fibers the model provides a tool for studying the roles of these nonlinearities in the encoding of simple and complex sounds in the responses of populations of an fibers

bounded support points for mappings with g parametric representation in c2 abstract in this paper we consider support points for the family s g 0 b 2 of mappings with g parametric representation on the euclidean unit ball b 2 in c 2 where g is a univalent function on the unit disc u in c which satisfies certain natural assumptions we shall use the shearing process recently introduced by bracci to prove the existence of bounded support points for the family s g 0 b 2 this result is in contrast to the one dimensional case where all support points of the family s are unbounded we also study the case of time log m reachable families r log m id b 2 m g generated by the caratheodory family m g and obtain certain results and applications which show a basic difference between the theory in the case of one complex variable and that in higher dimensions of particular interest is the case where g is a convex univalent function on u finally various consequences and certain conjectures are also considered

dispersive instability and its minimization in time domain computation of steady state responses of cochlear models dispersive instability appears in time domain solutions of classical cochlear models in this letter a derivation of optimal initial data is presented to minimize the effect of instability a second order accurate implicit boundary integral method is introduced numerical solutions of two dimensional models show that the optimal initial conditions work successfully in time domain steady state computations for both the zero neumann and zero dirichlet fluid pressure boundary conditions at the helicotrema

growth distortion and coefficient bounds for caratheodory families in cn and complex banach spaces abstract let x be a complex banach space with the unit ball b the family m is a natural generalization to complex banach spaces of the well known caratheodory family of functions with positive real part on the unit disc we consider subfamilies m g of m depending on a univalent function g we obtain growth theorems and coefficient bounds for holomorphic mappings in m g including some sharp improvements of existing results when g is convex we study the family r g consisting of holomorphic mappings f b x which have the property that the mapping d f z z belongs to m g further we consider radius problems related to the family r g when x is a complex hilbert space in particular if x is the euclidean space c n we obtain some quasiconformal extension results for mappings in r g we also obtain some sufficient conditions for univalence and starlikeness in complex banach spaces