pcr algorithm for the parallel computation of the solution of a class of singular linear systems abstract the paper presents a new highly parallel algorithm for computing the solution of a class of singular linear systems wawx b b r wa k 2 x r aw k 1 by the algorithm the solution x ad wb is obtained in t m 7 log 2 n 1 log 2 m 2 log 2 m r 1 log 2 mn steps with p mn processors when n 2 m 1 and with p 2m m 1 processors otherwise

solving constrained matrix equations and cramer rule this paper presents the solution of a general constrained matrix equation using generalized inverses and gives an explicit expression for the elements of the solution matrix using cramer rule

effect of variability in level on forward masking and on increment detection in the first of four experiments all with the same four subjects varying the level of a forward masker from interval to interval in a two interval forced choice 2ifc adaptive procedure had little effect on threshold in the second experiment the signal level was fixed and performance was measured in units of d varying the level of the forward masker again had little effect analyses of trial by trial data indicated that subjects did not vote for the interval with the higher level masker as would an energy detector performance was better on trials where the masker level in the interval with the signal was lower and was relatively independent of masker level in the nonsignal interval in the third experiment these results were replicated for a wider range of masker variability and with maskers lower in frequency than the signal in the fourth experiment the same range of variability from interval to interval was imposed on the level of the pedestal in an increment detection task results were sim

sources of variation in profile analysis ii component spacing dynamic changes and roving level profile analysis experiments have typically employed static profiles with constant frequency components spaced at equal intervals along a logarithmic frequency axis most periodic naturally occurring stimuli however have components that are harmonically related and vary dynamically in time one goal of these studies was to determine whether amplitude increment detection thresholds are different in dynamic harmonically spaced profiles compared to those for static log profiles and why such differences might exist a second goal was to determine the impact of roving levels within trial variation of level thresholds for static log profiles were on average 8 7 db lower than for static harmonic profiles a traditional filter bank model could not account for this result no consistent effect of dynamic contour an exponential rising frequency glide was observed thresholds were consistently poorer by 4 to 7 db when the level was roved but the differences in thresholds among the different profiles varied little it is proposed that the higher thresholds observed in static harmonic profiles may be accounted for by the more intense pitch strength associated with the harmonic profiles

a unified approach to krylov subspace methods for the drazin inverse solution of singular nonsymmetric linear systems abstract consider the linear system ax b where a c n n is a singular matrix in the present work we propose a general framework within which krylov subspace methods for drazin inverse solution of this system can be derived in a convenient way the krylov subspace methods known to us to date treat only the cases in which a is hermitian and its index ind a is unity necessarily in the present work a is not required to be hermitian it can have any type of spectrum and ind a is arbitrary we show that as is the case with nonsingular systems the krylov subspace methods developed here terminate in a finite number of steps that is at most n ind a for one of the methods derived here we also provide an analysis by which we are able to bound the errors the relevant bounds decreasing with increasing dimension of the krylov subspaces involved the results of this paper are applicable to consistent systems as well as to inconsistent ones an interesting feature of the approach to singular systems presented in this work is that it is formulated as a generalization of the standard krylov subspace approach to nonsingular systems indeed our approach here reduces to that relevant for nonsingular systems upon setting ind a 0 everywhere

spectral shape discrimination by hearing impaired and normal hearing listeners the ability to discriminate between sounds with different spectral shapes was evaluated for normal hearing and hearing impaired listeners listeners discriminated between a standard stimulus and a signal stimulus in which half of the standard components were decreased in level and half were increased in level in one condition the standard stimulus was the sum of six equal amplitude tones equal spl and in another the standard stimulus was the sum of six tones at equal sensation levels re audiometric thresholds for individual subjects equal sl spectral weights were estimated in conditions where the amplitudes of the individual tones were perturbed slightly on every presentation sensitivity was similar in all conditions for normal hearing and hearing impaired listeners the presence of perturbation and equal sl components increased thresholds for both groups but only small differences in weighting strategy were measured between the groups depending on whether the equal spl or equal sl condition was tested the average data suggest that normal hearing listeners may rely more on the central components of the spectrum whereas hearing impaired listeners may have been more likely to use the edges however individual weighting functions were quite variable especially for the hi listeners perhaps reflecting difficulty in processing changes in spectral shape due to hearing loss differences in weighting strategy without changes in sensitivity suggest that factors other than spectral weights such as internal noise or difficulty encoding a reference stimulus also may dominate performance

the effects of amplitude perturbation and increasing numbers of components in profile analysis in a profile analysis task the effect of randomly perturbing the amplitudes of the components of multi tone stimuli was studied in two experiments in the first experiment thresholds for a signal added in phase to the central component of a standard were measured for different numbers of components in two conditions in one condition thresholds were measured in blocks for six different jagged standards and in another thresholds were measured when one of the six standards was chosen randomly on a presentation by presentation basis regardless of condition thresholds did not depend on the numbers of components and increased magnitude of perturbation increased thresholds moreover the slope relating thresholds to number of components did not increase with increasing magnitude of perturbation in the second experiment the signal consisted of an increase in amplitude of the central components and a decrease in amplitude of the outer components of the standard a stimulus type which has been shown to maximize the change in threshold with increasing number of components the amplitudes of component tones were selected randomly on a presentation by presentation basis thresholds fell with increases in the number of components but the slope relating thresholds to numbers of components did not change as the magnitude of perturbation increased the latter result contrasts with that reported by kidd et al j acoust soc am 90 1340 1354 1991

variation in spectral shape discrimination weighting functions at different stimulus levels and signal strengths this study evaluated whether weights for spectral shape discrimination depend on overall stimulus level and signal strength the degree of spectral shape change between two stimuli five listeners discriminated between standard stimuli that were the sum of six equal amplitude tones and signal stimuli created by decreasing the amplitudes of three low frequency components and increasing the amplitudes of three high frequency components weighting functions were influenced by stimulus level in that the relative contribution of the low frequency decremented components to the high frequency incremented components decreased with increasing stimulus level although individual variability was present a follow up experiment suggested that the level dependence was due to greater reliance on high frequency components rather than incremented components excitation pattern analyses indicated that the level dependence is primarily but not solely driven by cochlear factors in general different signal strengths had no effect on the weighting functions when normalized but two of the five listeners showed variability in the shape of the weighting function across signal strengths results suggest that the effects of stimulus level on weighting functions and individual variability in the shapes of the weighting functions should be considered when comparing weighting functions across conditions and groups that might require different stimulus levels and signal strengths

detectability of increments and decrements in spectral profiles in auditory profile analysis the subject typically has to discriminate a standard spectrum consisting of widely spaced equal amplitude sinusoids from a comparison in which one or several components have been elevated relative to the others according to the widely accepted multichannel model however it should not matter whether the spectral shape alteration consists of incrementing or decrementing a component by a small number of decibels thus in order to investigate the generality of the hypothesized mechanism and to explore the role of pitch salience cues which might be thought to favor increments to single components two experiments comparing the detectability of intensity increments and decrements in spectral profiles were conducted in experiment 1 in which the number of components was varied from 3 to 15 within a fixed frequency range decrements to the central 800 hz component produced significantly larger thresholds than did increments experiment 2 which employed three tone complexes spanning different frequency ranges revealed that this effect is entirely due to the spectral density of the frequency components as the distance between the variable signal component and the flanking reference sinusoids decreased the increment decrement disparity became larger these effects may at least qualitatively be accounted for when peripheral excitation patterns are considered suggesting that local decrements are constrained more than increments by the excitation intruding from flanking stimulus components

perceptual weights in auditory level discrimination perceptual weights in level discrimination also called intensity discrimination were determined for 3 7 15 and 24 component tone complexes with flat spectral envelopes using a correlational paradigm each frequency component was randomly and independently perturbed in level on each presentation for the target interval frequency component levels were additionally increased by the level increment to be detected l 20 log10 p p p where p is pressure weights were calculated from the across trial correlation between the level perturbations for each frequency component and the interval chosen by the listener two conditions were investigated 1 l was equal across frequency components and 2 l increased progressively across frequency components for both conditions data for four listeners usually showed the greatest weight for the highest frequency component the two to four highest frequency components generally were most important for level discrimination the effect of increasing l

measuring decision weights in recognition experiments with multiple response alternatives comparing the correlation and multinomial logistic regression methods psychophysical reverse correlation methods allow researchers to gain insight into the perceptual representations and decision weighting strategies of individual subjects in perceptual tasks although these methods have gained momentum until recently their development was limited to experiments involving only two response categories recently two approaches for estimating decision weights in m alternative experiments have been put forward one approach extends the two category correlation method to m 2 alternatives the second uses multinomial logistic regression mlr in this article the relative merits of the two methods are discussed and the issues of convergence and statistical efficiency of the methods are evaluated quantitatively using monte carlo simulations the results indicate that for a range of values of the number of trials the estimated weighting patterns are closer to their asymptotic values for the correlation method than for the mlr method moreover for the mlr method weight estimates for different stimulus components can exhibit strong correlations making the analysis and interpretation of measured weighting patterns less straightforward than for the correlation method these and other advantages of the correlation method which include computational simplicity and a close relationship to other well established psychophysical reverse correlation methods make it an attractive tool to uncover decision strategies in m alternative experiments

the measurement problem in level discrimination there is disagreement among theorists over the exact measure to be used to quantify auditory level discrimination it has been proposed that for level discrimination tasks the measure that is most linearly related to the sensitivity index d will be the correct measure the level difference l and the weber fraction are both candidates though the latter is sensitive to the physical unit in which it is expressed e g pressure or intensity while the former is not psychometric functions for level discrimination were obtained at a number of pedestal levels for 10 ms sinusoids either 1000 or 6500hz and broadband noise bursts these functions were used to assess which of three measures l p p or i i is most nearly linearly related to d the results suggest that p p is the measure that comes closest to being linearly related to d

analytical calculation of two leading exponents of the dilute potts model a potts model on a square lattice with two and four spin interaction and site and bond dilution is shown to be dual to itself the model is mapped onto a vertex problem which in turn is equivalent to a solid on solid model by means of these mappings the dilute potts model can be written as a gaussian like model with staggered and direct periodic fields the leading and next to leading exponents of the potts model are calculated subject to the validity of certain assumptions

intensity weighted average of instantaneous frequency as a model for frequency discrimination the intensity weighted average of instantaneous frequency iwaif is developed as a model to predict listener performance in tasks primarily requiring frequency discrimination iwaif is closely related to the envelope weighted average of instantaneous frequency ewaif model proposed by feth for similar tasks the primary difference is that the iwaif model uses intensity envelope squared as the weighting function instead of the envelope the advantages of iwaif over ewaif are that a it has a convenient frequency domain interpretation and b it is much simpler to compute than the ewaif the iwaif is the center of gravity of the energy spectral density function of the signal

masker fluctuation and the masking level difference previous work suggests that for some listeners masker fluctuation may be advantageous for nos detection this study tested the hypothesis that the benefit of masker fluctuation to binaural analysis is based on the ability to take advantage of epochs of low masker energy in the fluctuating masker where the cues underlying binaural signal detection are more salient the hypothesis was evaluated using the time domain version of coss analysis derived by buus et al j acoust soc am 99 2288 2297 1996 which measures the perceptual weight applied by a listener within a relatively brief time window as a function of the masker level during the window the results indicated a dependency of signal detection on short term masker level in the nos condition but not in the noso condition this finding supports a new perspective indicating that binaural signal detection depends upon the envelope of a masker in a way that is fundamentally different from that typically associated with monaural detection

some additive results on drazin inverses in this paper some additive results on drazin inverse of a sum of drazin invertible elements are derived some converse results are also presented

detection of missing modal frequencies the amount of information about a source that can be recovered from sound is naturally limited by the ear s ability to resolve individual modal frequencies unique to that source to measure these limits listeners were asked to detect in a standard two interval forced choice task with feedback which of two sounds on each trial contained a missing partial the frequencies of the partials corresponded to the ideal modes of a simply supported rectangular plate plate surface areas and height wide ratios were chosen to produce the same bandwidths 125 1125 250 2250 or 500 4500 hz for different numbers of partials 11 16 or 24 overall level of the sounds was roved to discourage detection based on simple level differences detection of five highly practiced listeners was largely independent of the frequency of the lowest partial being best for partials 1 3 1 4 or 1 6 for 11 16 or 24 partials in total respectively mutual masking among the higher number partials is given as the likely cause the res

different auditory filter bandwidth estimates based on profile analysis notched noise and hybrid tasks auditory filter bandwidths were estimated in three experiments the first experiment was a profile analysis experiment the stimuli were composed of sinusoidal components ranging in frequency from 200 to 5000 hz the standard stimulus was the sum of equal amplitude tones and the signal stimulus had a power spectrum that varied up down up down the number of components ranged from four to 60 interval by interval level randomization prevented the change in level of a single component from reliably indicating the change from standard to signal the second experiment was a notched noise experiment in which the 1000 hz tone to be detected was added to a noise with a notch arithmetically centered at 1000 hz detection thresholds were estimated both in the presence of and in the absence of level randomization in the third hybrid experiment a 1000 hz tone was to be detected and the masker was composed of equal amplitude sinusoidal components ranging in frequency from 200 to 5000 hz for this experiment thr

nonmonotonicity of informational masking neff and green percept psychophys 41 409 415 1987 report that the masking of single tones by random frequency multitone maskers varies nonmonotonically with number of masker components peaking at 10 50 components in this paper it is shown that such results are well predicted by a model the component relative entropy model core wherein thresholds increase linearly with the ensemble variance of masker spectra smoothed by peripheral auditory filters r a lutfi j acoust soc am 94 748 758 1993 three experiments were conducted in the first the nonmonotonic relation was replicated for 9 of 11 listeners in conditions similar to those of neff and green in the second the frequencies of masker components were fixed and the levels of components were varied randomly across presentations to simulate gaussian noise in this case the nonmonotonicity and the total amount of masking for these listeners were shown to be significantly reduced in the third experiment masked thresholds for the signal were found to vary monotonically with the frequency spacing of masker components for a fixed number of masker components large individual differences among listeners were obtained in some experimental conditions individual as well as mean thresholds were well predicted by the core model with an appropriate selection of the values of the two free parameters of the model for each listener

evaluation of simple models of auditory profile analysis using random reference spectra this article describes further study of the finding reported by green et al j acoust soc am 73 639 643 1983 and others that in certain conditions the threshold of detectability for an intensity increment to the center tone of a multitone reference spectrum decreased as the number of nonsignal tones increased that result was considered remarkable since critical band theory would predict that these nonsignal tones spaced outside the critical band containing the signal would have no effect on or at most slightly decrease within band detectability and certainly could not account for the result of improved detectability found in the study cited above recently henn and turner j acoust soc am 88 126 131 1990 were unable to replicate the result described above concluding that the phenomenon exists only in limited conditions and that it is highly individual in nature further they speculated that the most likely reason for the discrepancy between their study and previous s

determinantal representations of the w weighted drazin inverse over the quaternion skew field within the framework of the theory of the column and row determinants we obtain new determinantal representations of the w weighted drazin inverse over the quaternion skew field we give determinantal representations of the w weighted drazin inverse by using previously introduced determinantal representations of the drazin inverse the moore penrose inverse and the limit representations of the w weighted drazin inverse in some special case

static and dynamic critical phenomena of the two dimensional q state potts model theq state potts model on the square lattice is studied by monte carlo simulation forq 3 4 5 6 very good agreement is obtained with exact results of kiharaet al and baxter for energy and free energy at the critical point critical exponent estimates forq 3 are 0 4 0 1 1 45 in rough agreement with high temperature series extrapolation and real space renormalization group methods the transition forq 5 6 is found to be a very weakly first order transition i e pronounced pseudocritical phenomena occur specific heat susceptibility etc nearly diverge at the first order transition temperature dynamics is associated to the model in the same way as for the kinetic ising model and the nonlinear slowing down of the order parameter and of the energy is studied the dynamic exponent is estimated to be zv 1 9 within our accuracy noq dependence is detected the relaxation is found to be consistent with dynamic scaling predictions and dynamic scaling functions associated with the nonlinear relaxation are estimated

spectral peak selection in spectral shape discrimination by normal hearing and hearing impaired listeners spectral shape discrimination thresholds were measured in the presence and absence of noise to determine whether normal hearing and hearing impaired listeners rely primarily on spectral peaks in the excitation pattern when discriminating between stimuli with different spectral shapes standard stimuli were the sum of 2 4 6 8 10 20 or 30 equal amplitude tones with frequencies fixed between 200 and 4000hz signal stimuli were generated by increasing and decreasing the levels of every other standard component the function relating the spectral shape discrimination threshold to the number of components n showed an initial decrease in threshold with increasing n and then an increase in threshold when the number of components reached 10 and 6 for normal hearing and hearing impaired listeners respectively the presence of a 50 db spl hz noise led to a 1 7db increase in threshold for normal hearing listeners and a 3 5db increase for hearing impaired listeners multichannel modeling and the relatively s

signal uncertainty and psychometric functions in profile analysis experiment 1 was conducted to compare the effects of signal frequency uncertainty on the detection of a change in spectral shape and on the detection of a tone in wideband noise results indicate that for both tasks the uncertainty effect was small being on average about 3 db in a second experiment psychometric functions were measured for the detection of changes in the spectral shape of multicomponent complexes psychometric functions for profile tasks have a 25 db range and are similar to those measured for the detection of an increment in the level of a single sinusoid these psychometric functions are different from those found when detecting a signal in noise which typically have a 10 db range three equations for the shape of the psychometric functions were compared the difference in the resulting fits was small thus preventing an unambiguous choice of functional form

investigation of perceptual constancy in the temporal envelope domain the ability to discriminate complex temporal envelope patterns submitted to temporal compression or expansion was assessed in normal hearing listeners an xab matching to sample procedure was used x the reference stimulus is obtained by applying the sum of two inharmonically related sinusoids to a broadband noise carrier a and b are obtained by multiplying the frequency of each modulation component of x by the same time expansion compression factor 0 35 2 83 for each trial a or b is a time reversed rendering of x and the listeners task is to choose which of the two is matched by x overall the results indicate that discrimination performance degrades for increasing amounts of time expansion compression i e when departs from 1 regardless of the frequency spacing of modulation components and the peak to trough ratio of the complex envelopes an auditory model based on envelope extraction followed by a memory limited template matching process accounted for results obtained withou

informational masking with small set sizes informational masking refers to interference in the detectability of a sound or discrimination of some property of a sound beyond that which can be attributed to interactions at the auditory periphery in the current experiments the signal to be detected was a tone added to a 6 tone masker and informational masking was introduced by randomly choosing the frequencies of the tones that comprise the masker the primary question was whether small numbers of maskers could replace randomly drawn maskers without sacrificing the underlying detection schemes adopted by observers similar to the method used by wright and saberi j acoust soc am 105 1765 1775 1999 detection thresholds were measured for different masker set sizes where set size refers to the number of 6 tone maskers from which any one masker was drawn set sizes of 3 6 12 and 24 were tested as well as conditions in which the maskers were chosen at random in addition observers memory for maskers was coarsely evaluated large differences in thresholds were found across observers and across different masker sets even for set sizes of 24 the memory test suggests some recognition of maskers for some observers post hoc analysis of the data included an evaluation of the relative contribution of different frequencies using a single linear model as a base for comparison a linear model fitted to each condition was also evaluated although the data were fitted better using many rather than one linear model the reduction in quality of fit was modest this result suggests substantial consistency in decision strategies regardless of masker set size

on the symmetric solutions of linear matrix equations abstract the symmetric solutions of linear matrix equations ax c and axb c are considered the necessary and sufficient conditions for the consistency of the equations with a symmetric condition on solutions are derived using the singular value decomposition and the generalized singular value decomposition numerical algorithms for solving the equations are suggested

dilute potts model duality and site bond percolation a number of results are obtained for potts models on an arbitrary planar lattice it is shown that the dual of a dilute potts model is a graph generating function and that the dual of a constrained dilute potts model is an undiluted potts model it is also shown that the dilute potts model generates a correlated site bond percolation for uncorrelated site bond percolations the analysis determines the percolation threshold from a knowledge of the critical point of a potts model this generalises a recent result of kondor 1980 who uses a star triangle transformation to derive this relationship for the honeycomb and triangle lattices

a molecular description of profile analysis decision weights and internal noise systematic inefficiencies and internal noise in a spectral profile discrimination task were investigated listeners detected a 1000 hz sinusoid added in phase to the central component of a complex consisting of 11 equal intensity sinusoids parameters for a channel model that employs decision weights and internal noise were estimated with molecular psychophysical techniques maximum likelihood predictions of the model were generally within a few decibels of observed thresholds the degree to which an assumption of ideal weights leads to overestimation of internal noise was also assessed

auditory profile analysis of irregular sound spectra the discrimination of changes in the shapes of sound spectra is reported the change was always an intensity increment to the 948 hz component of a multitone complex first the ability of naive listeners to learn to discriminate a change in a regular background or reference spectrum equal level tones equally spaced in logarithmic frequency was measured as a function of the number of trials on the average threshold improved about 10 db over 3000 trials with about 50 of the decrease in threshold occurring during the first 750 trials in a subsequent series of experiments the overall pattern of spectral shape of the background was varied randomly two kinds of perturbations in spectral shape were employed 1 randomly choosing the frequencies of the reference spectra and 2 randomly choosing the amplitudes of the components of the reference spectra the experimental manipulations involved fixing the random spectra across a block of trials varying the reference spectra from interval to interval of each trial and providing extensive practice in discriminating specific randomly perturbed reference spectra the results of the spectrum learning and random perturbation experiments provide insight into the roles of critical band filtering sensory variability and short term and long term memory representations in auditory profile analysis further the appropriateness of the generalization of a simple energy detection model is discussed

effect of temporal position proportional variance and proportional duration on decision weights in temporal pattern discrimination two experiments investigated how listeners allocate their attention to different segments of a temporal pattern the experiments allowed a direct test of the predictions of the proportion of total duration ptd rule and the component relative entropy core model listeners had to decide whether two sequences of nine tones had the same or different temporal patterns tone duration 25 ms tone frequency 1000 hz a sequence s temporal pattern was determined by the time intervals between each tone s offset and the next tone s onset on same trials the time intervals at corresponding temporal positions in the two sequences were identical on different trials the corresponding time intervals were randomly varied listener attention to different temporal positions within a sequence was assessed by calculating the decision weights at each position the results supported the core model and were inconsistent with the ptd rule manipulating the mean of the time intervals within the sequence had no consistent ef

condensed cramer rule for solving restricted matrix equations a cramer rule for solving a kind of restricted matrix equationswawxw bw d r x r aw k 1 n x n w b k 2 is presented in g wang jie sun a cramer rule for solution of the general restricted matrix equation appl math comput 154 2004 415 422 this paper gives a more condensed cramer rule for the solution of the matrix equations the results in g wang z l xu solving a kind of restricted matrix equations and cramer rule appl math comput 162 2005 329 338 chao gu g r wang z l xu pcr algorithm for the parallel computation of the solution of a class of singular linear systems appl math comput 176 2006 237 244 are partially the special cases in our paper

explicit formulas for determinantal representations of the drazin inverse solutions of some matrix and differential matrix equations the drazin inverse solutions of the matrix equations ax b xa b and axb d are considered in this paper we use both the determinantal representations of the drazin inverse obtained earlier by the author and in the paper we get analogs of the cramer rule for the drazin inverse solutions of these matrix equations and using their for determinantal representations of solutions of some differential matrix equations x ax b and x xa b where the matrix a is singular

determinantal representations of the drazin inverse over the quaternion skew field with applications to some matrix equations abstract within the framework of the theory of the column and row determinants we obtain determinantal representations of the drazin inverse both for hermitian and arbitrary matrices over the quaternion skew field using the obtained determinantal representations of the drazin inverse we get explicit representation formulas analogs of cramer s rule for the drazin inverse solutions of a quaternion matrix equation axb d and consequently ax d and xb d in two cases if a b are hermitian or arbitrary

solving a kind of restricted matrix equations and cramer rule the problem of finding determinantal formula for solutions of some restricted linear systems ax b has been discussed in linear multilinear algebra 15 1984 319 linear algebra appl 116 1989 27 linear multilinear algebra 34 1993 177 appl math comput 125 2002 303 this paper deals with some more extensive cases of this kind of problem and establishes determinantal formulas for solutions of the restricted matrix equationsax d r x r a k 1 xb d n x n b k 2 axb d r x r a k 1 n x n b k 2 where a c n x n with ind a k 1 b c m x m with ind b k 2 and d c n x m the results in linear multilinear algebra 15 1984 319 linear algebra appl 74 1986 213 linear multilinear algebra 34 1993 177 appl math comput 125 2002 303 are partially the special cases in our paper the classic cramer rule is also a special case of our results

an iterative method for solving general restricted linear equations the newton iterative method for computing outer inverses with prescribed range and null space is used in the non stationary richardson iterative method to develop an iterative method for solving general restricted linear equations starting with any suitably chosen initial iterate our method generates a sequence of iterates converging to the solution the necessary and sufficient conditions for the convergence along with the error bounds are established the applications of the iterative method for solving some special linear equations are also discussed a number of numerical examples are worked out they include singular square rectangular randomly generated rank deficient matrices full rank matrices and a set of singular matrices given in matrix computation toolbox mctoolbox with the condition numbers ranging from order 1016 to 1050 the mean cpu time mct and the error bounds are the performance measures used our results when compared with the results obtained by chen 1997 leads to substantial improvement in terms of both computational speed and accuracy

the number of components in profile analysis tasks the central issue of this paper is how detection of a change in spectral shape depends on the number of components m that define the spectrum first a channel theory is reviewed and the predictions relating the detectability of such a change as a function of the number of components are derived second the existing literature is reviewed most of it concerns the detection of an increment in a single tone of an m component equal amplitude complex the data are close 2 db to the predictions of the theory which suggests that there is a relatively small change in threshold as the number of components increases the consistency of the data is also discussed and hypotheses for discrepancies are presented finally the results of an experiment measuring the detectability of a special spectral shape change is explored in this case all the changes are equal in size but some are opposite in sign so that the summed change is zero the results using this balanced spectral change are also consistent with the predictions of the theory and show that for this special kind of spectral change thresholds improve by nearly 7 db as m increases from 4 to 24 components

ranks of least squares solutions of the matrix equation axb c for a complex matrix equation axb c we solve the following two problems 1 the maximal and minimal ranks of least square solution x to axb c and 2 the maximal and minimal ranks of two real matrices x 0 and x 1 in least square solution x x 0 ix 1 to axb c we also give a necessary and sufficient condition for matrix equations a ix ib i c i i 1 2 to have a common least square solution

analogs of cramer s rule for the minimum norm least squares solutions of some matrix equations abstract the least squares solutions with the minimum norm of the matrix equations ax b xa b and axb d are considered in this paper we use the determinantal representations of the moore penrose inverse obtained earlier by the author and get analogs of the cramer rule for the minimum norm least squares solutions of these matrix equations

psychometric functions for pure tone frequency discrimination the form of the psychometric function pf for auditory frequency discrimination is of theoretical interest and practical importance in this study pfs for pure tone frequency discrimination were measured for several standard frequencies 200 8000 hz and levels 35 85 db sound pressure level spl in normal hearing listeners the proportion correct data were fitted using a cumulative gaussian function of the sensitivity index d computed as a power transformation of the frequency difference f the exponent of the power function corresponded to the slope of the pf on log d log f coordinates the influence of attentional lapses on pf slope estimates was investigated when attentional lapses were not taken into account the estimated pf slopes on log d log f coordinates were found to be significantly lower than 1 suggesting a nonlinear relationship between d and f however when lapse rate was included as a free parameter in the fits pf slopes were found not to differ significantly from 1 consistent with a linear relationship between d and f this was the case across the wide ranges of frequencies and levels tested in this study therefore spectral and temporal models of frequency discrimination must account for a linear relationship between d and f across a wide range of frequencies and levels

level discrimination of single tones in a multitone complex by normal hearing and hearing impaired listeners a conditional on a single stimulus coss analysis procedure j acoust soc am 86 1743 1746 1989 was used to estimate how well normal hearing and hearing impaired listeners selectively attend to individual spectral components of a broadband signal in a level discrimination task on each trial two multitone complexes consisting of six octave frequencies from 250 to 8000 hz were presented to listeners the levels of the individual tones were chosen independently and at random on each presentation the target tone was selected within a block of trials as the 250 1000 or 4000 hz component on each trial listeners were asked to indicate which of the two complex sounds contained the higher level target as a group normal hearing listeners exhibited greater selectivity than hearing impaired listeners to the 250 hz target while hearing impaired listeners showed greater selectivity than normal hearing listeners to the 4000 hz target which is in the region of their hearing loss both groups of listeners displayed large variability in their ability to selectively weight the 1000 hz target trial by trial analysis showed a decrease in weighting efficiency with increasing frequency for normal hearing listeners but a relatively constant weighting efficiency across frequency for hearing impaired listeners interestingly hearing impaired listeners selectively weighted the 4000 hz target which was in the region of their hearing loss more efficiently than did the normal hearing listeners

spectral weights in profile listening the coss analysis suggested by berg b g berg j acoust soc am 86 1743 1746 1989 is applied to a profile listening task the listener s task is to detect an increment in the level of the middle component of an n component spectrum the overall level of the components is randomly selected from a 20 db range on each presentation thus the detection task is essentially one of detecting a change in spectral shape to implement the coss analysis a small perturbation in level is added to each component of the complex coss functions are generated from these perturbations and the spectral weight that the listener assigns to each component is estimated data are reported for n 3 5 and 11 components and for perturbations with standard deviations of 0 5 1 and 2 db the estimated weights are similar to those derived for an optimum detector namely the level at the signal component is compared with the average level of the nonsignal components this result supports the view that profile analysis invol

detection of time and bandlimited increments and decrements in a random level noise the purpose of this study was to compare detection of increments and decrements occurring over limited regions of time and frequency within a 500 ms broadband 0 6000 hz noise three listeners tracked detection thresholds adaptively in a two interval two alternative forced choice task thresholds were measured for both increments and decrements in level l 10 log10 1 n0 n0 db where n0 is the spectral power density of the noise as a function of signal duration t 30 500 ms for a range of signal bandwidths w 62 6000 hz that were logarithmically centered around 2500 hz listeners were forced to rely on temporal and spectral profile cues for detection due to randomization of overall presentation level from interval to interval which rendered overall energy an inconsistent cue increments were detectable for all combinations of w and t whereas decrements were not consistently detectable for w 500 hz narrow band decrements were not detectable due to spread of excitation from the spectral edges of

on the minimum audible difference in direct to reverberant energy ratio the goals of this study were to measure sensitivity to the direct to reverberant energy ratio d r across a wide range of d r values and to gain insight into which cues are used in the discrimination process the main finding is that changes in d r are discriminated primarily based on spectral cues temporal cues may be used but only when spectral cues are diminished or not available while sensitivity to interaural cross correlation is too low to be useful in any of the conditions tested these findings are based on an acoustic analysis of these variables and the results of two psychophysical experiments the first experiment employs wideband noise with two values for onset and offset times to determine the d r just noticeable difference at 10 0 10 and 20db d r this yielded substantially higher sensitivity to d r at 0 and 10db d r 2 3db than has been reported previously while sensitivity is much lower at 10 and 20db d r the second experiment consists of three parts where specific cues to d r are reduced or removed which enabled the specified rank ordering of the cues the acoustic analysis and psychophysical experiments also provide an explanation for the auditory horizon effect

recurrent neural network for computing the w weighted drazin inverse abstract two gradient based recurrent neural networks rnns for computing the w weighted drazin inverse of a real rectangular matrix are proposed and considered usage of the first rnn is limited by a specific constraint on the spectrum of a certain matrix the second rnn is usable without restrictions the stability of the recurrent neural networks as well as their convergence are considered numerical examples are given to show the efficiency of the proposed neural networks

decision strategies of hearing impaired listeners in spectral shape discrimination the ability to discriminate between sounds with different spectral shapes was evaluated for normal hearing and hearing impaired listeners listeners detected a 920 hz tone added in phase to a single component of a standard consisting of the sum of five tones spaced equally on a logarithmic frequency scale ranging from 200 to 4200 hz an overall level randomization of 10 db was either present or absent in one subset of conditions the no perturbation conditions the standard stimulus was the sum of equal amplitude tones in the perturbation conditions the amplitudes of the components within a stimulus were randomly altered on every presentation for both perturbation and no perturbation conditions thresholds for the detection of the 920 hz tone were measured to compare sensitivity to changes in spectral shape between normal hearing and hearing impaired listeners to assess whether hearing impaired listeners relied on different regions of the spectrum to discriminate between sounds spectral weights were estimated from the perturbed standards by correlating the listener s responses with the level differences per component across two intervals of a two alternative forced choice task results showed that hearing impaired and normal hearing listeners had similar sensitivity to changes in spectral shape on average across frequency correlation functions also were similar for both groups of listeners suggesting that as long as all components are audible and well separated in frequency hearing impaired listeners can use information across frequency as well as normal hearing listeners analysis of the individual data revealed however that normal hearing listeners may be better able to adopt optimal weighting schemes this conclusion is only tentative as differences in internal noise may need to be considered to interpret the results obtained from weighting studies between normal hearing and hearing impaired listeners

algebraic properties of the rank deficient equality constrained and weighted least squares problems abstract we study the algebraic properties of the solutions of the equality constrained least squares problem min f s v kf g v 2 with s f v h lf v 2 min z c n v v h lz v 2 lse and corresponding weighted least squares problem wls in which l and l k may not be of full rank general formulas for the solutions are given and the algebraic relations between the lse and the wls solutions are obtained

cuing effects for informational masking the detection of a tone added to a random frequency multitone masker can be very poor even when the maskers have little energy in the frequency region of the signal this paper examines the effects of adding a pretrial cue to reduce uncertainty for the masker or the signal the first two experiments examined the effect of cuing a fixed frequency signal as the number of masker components and presentation methods were manipulated cue effectiveness varied across observers but could reduce thresholds by as much as 20 db procedural comparisons indicated observers benefited more from having two masker samples to compare with or without a signal cue than having a single interval with one masker sample and a signal cue the third experiment used random frequency signals and compared no cue signal cue and masker cue conditions and also systematically varied the time interval between cue offset and trial onset thresholds with a cued random frequency signal remained higher than for a cued fixed frequency signal for time intervals between the cue and trial of 50 ms or longer thresholds were approximately the same with a signal or a masker cue and lower than when there was no cue without a cue or with a masker cue analyses of possible decision strategies suggested observers attended to the potential signal frequencies particularly the highest signal frequency with a signal cue observers appeared to attend to the frequency of the subsequent signal

a cramer rule for finding the solution of a class of singular equations a cramer rule for finding the unique solution x r ak of singular equations ax b ind a k b r ak is given and reduces to the common cramer rule if a is a nonsingular

on the symmetric solutions of a linear matrix equation a formula for the partitioned minimum norm reflexive generalized inverse is applied to find the general symmetric solution x to the matrix equation ax b also the dimension of the space of symmetric solutions is established

spectrographic measurement of instantaneous frequency and the time dependent weighted average instantaneous frequency instantaneous frequency if is an important signal characteristic arising in many fields it is a concept intimately linked to time frequency analysis where it can be obtained from a time frequency distribution tfd as the first conditional moment in frequency suggesting that the if is the average frequency at each time however this interpretation is questionable since it is well known that the if often ranges beyond the spectral support of the signal in addition to obtain the if from a spectrogram which is one possible tfd a very wideband and thus severely spectrally distorted spectrogram must be used more reasonable bandwidths are investigated and in particular give the conditions by which the first conditional moment of a spectrogram can indeed be interpreted as the average frequency at each time under these conditions namely when the spectrogram bandwidth is less than the frequency separation of the signal components in the time frequency plane the spectrogram yields not the usual if but a time dependent weighted average instantaneous frequency waif of the signal also while the if and waif are generally different sometimes they are the same in particular when there is symmetry in the time frequency spectrum of the signal in that case the first conditional spectral moments of both wideband and narrow band spectrograms are the same and interpretable as the average frequency at each time

sensitivity to changes in overall level and spectral shape an evaluation of a channel model two experiments involving level and spectral shape discrimination which test an optimal channel model developed by durlach et al j acoust soc am 80 63 72 1986 are described the model specifies how the auditory system compares and or combines intensity information in different frequency channels in the first experiment psychometric functions were obtained for the discrimination of changes in level and discrimination of changes in spectral shape for an eight tone complex sound a variety of different base spectral shapes were tested in some conditions level randomization was introduced to reduce the reliability of across interval changes in level increasing the amount of level variation degraded performance for the level discrimination task but had no effect on the shape discrimination task in all conditions sensitivity to changes in spectral shape was superior to sensitivity to changes in level consequently two models of central noise are evaluated in an attempt to explain these results one in which central noise acts prior to the formation of the likelihood ratio and one in which central noise degrades the likelihood ratio the former model is more successful in accounting for the data in a second experiment the detectability of a level increment to one component of a multitone complex was measured the frequency content of the complex was varied by systematically removing six components from a 23 component complex thresholds were measured for increments at three different signal frequencies a common trend in the data was that when there was a spectral gap directly above the signal frequency thresholds were lowest this result differs from the predictions of a simple channel model and contrasts with results presented by green and berg q j exp psychol 43a 449 458 1991

auditory profile analysis is there perceptual constancy for spectral shape for stimuli roved in frequency the ability to discriminate between two spectral shapes was investigated using a one interval two alternative forced choice procedure each stimulus consisted of seven components equally spaced on a logarithmic frequency scale frequency ratio of 1 2 between adjacent components the two spectral shapes were produced by first generating a complex each of whose components was equally loud when heard in isolation and then increasing the level of either the third or the fifth component by a certain amount to render pitch an unusable cue the center frequencies of the stimuli were randomly varied between presentations maintaining the ratio between adjacent components over a range of about three octaves 279 2074 hz the data indicate that listeners were not able to distinguish between the two types of profiles even for very large increments in level of the third and the fifth components the results suggest that there is no perceptual constancy for this type of spectral shape for stimuli roved over a sufficiently large frequency range the spectral shape cannot be perceived independently of stimulus frequency the results are related to the multi channel model applied in profile analysis tasks

sources of variation in profile analysis i individual differences and extended training this study investigated two sources of variance in the ability to discriminate auditory profiles individual differences and extended training the goals of the study were 1 to determine the range and origins of individual differences in profile analysis and 2 to determine whether those who initially had poor sensitivity to changes in spectral shape could eventually acquire finer sensitivity profile stimuli had 11 components with equal log spacing from 200 2200 hz thresholds ranged from 1 to 25 db signal level relative to the context level across 46 listeners the correlation between spectral shape discrimination thresholds after 2000 trials and pure tone intensity discrimination thresholds was 0 36 the range of individual differences for pure tone intensity discrimination and spectral shape discrimination was about the same two groups of listeners were given extended practice on the profile task one group that showed low thresholds after an initial 2000 trials of practice and another that showed much higher initial thresholds all listeners improved during the course of the first 2000 trials of training most of the poor listeners continued to improve during 9000 trials of training individual differences in the listeners sensitivity to changes in spectral shape still existed after the extended practice

molecular analysis of the effect of relative tone level on multitone pattern discrimination molecular psychophysics attempts to model the observer s response to stimuli as they vary from trial to trial the approach has gained popularity in multitone pattern discrimination studies as a means of estimating the relative reliance or decision weight listeners give to different tones in the pattern various factors affecting decision weights have been examined but one largely ignored is the relative level of tones in the pattern in the present study listeners detected a level increment in a sequence of 5 100 ms 2 0 khz tone bursts alternating in level between 40 and 80db spl the level increment was made largest on the 40 db tones yet despite this all four highly practiced listeners gave near exclusive weight to the 80 db tones the effect was the same when the tones were replaced by bursts of broadband gaussian noise alternating in level it was reduced only when the level differences were made 10db and it was entirely reversed only when the low level tones alternated with louder bursts of ga

successive versus simultaneous comparison in auditory intensity discrimination the ability to discriminate between two brief sounds having identical frequency components but differing in the intensity of one or more of the components is studied the stimulus manipulations include randomizing the overall intensity of the sounds varying the number and spacing of the components and varying the interval of time between the sounds the results from the experiment in which interstimulus interval is manipulated clearly support a profile analysis mechanism that computes two rough spectral analyses stores a classification of these spectra in memory and compares the two stores this profile mechanism involves a simultaneous comparison of different components of the complex some remote from the signal frequency rather than a successive comparison of the difference in intensity at the signal frequency the simultaneous comparison process is more sensitive when the profile is composed of many components spaced over a wide frequency range

multicritical behaviour in the q state potts lattice gas the phase diagram of the q state potts lattice gas is determined exactly on a cayley tree for q 2 the authors use this result to make some conjectures concerning the lower dimensional cases it is argued that there should be a tetracritical point different from the transition point where ising droplets diverge in the case of three dimensional correlated site bond percolation q 1 in two dimensions the critical exponent gamma p of the mean cluster size at the ising critical point is derived from the extended den nijs conjectures gamma p 91 48 1 896

a characterization for the w weighted drazin inverse and a cramer rule for the w weighted drazin inverse solution we establish characterization for the w weighted drazin inverse of an arbitrary rectangular matrix which reduces to the well known result if the matrix is nonsingular also a cramer rule for finding the unique w weighted drazin inverse solution x r aw k 1 of special restricted linear equationswawx b b r wa k 2 is presented and reduces to the classical cramer rule if a is invertible

signal frequency uncertainty in spectral shape discrimination psychometric functions psychometric functions were measured for spectral shape discrimination profile analysis with the signal frequency either fixed what we call the signal known condition or randomly varied what we call the signal unknown condition the functions were obtained using an adaptive up down procedure in the signal unknown condition independent tracks for each signal frequency were interleaved within the same block of trials the mean slope of the psychometric function k in log d k log delta l c where delta l 20 log 1 delta p p was 1 13 for the signal known conditions and 1 19 for the signal unknown condition the mean signal to standard ratio at threshold pc 79 4 obtained in the signal unknown condition was 3 to 4 db higher than that obtained in the signal known conditions the psychometric functions of the ideal observer were derived for both the signal known and signal unknown conditions a comparison of the measured and derived psychometric functions suggests that the internal noise component that is statistically independent across frequency channels contributes minimally to the total internal and external noise sources that are involved in the decision process

detection of random alterations to time varying musical instrument spectra the time varying spectra of eight musical instrument sounds were randomly altered by a time invariant process to determine how detection of spectral alteration varies with degree of alteration instrument musical experience and spectral variation sounds were resynthesized with centroids equalized to the original sounds with frequencies harmonically flattened and with average spectral error levels of 8 16 24 32 and 48 listeners were asked to discriminate the randomly altered sounds from reference sounds resynthesized from the original data for all eight instruments discrimination was very good for the 32 and 48 error levels moderate for the 16 and 24 error levels and poor for the 8 error levels when the error levels were 16 24 and 32 the scores of musically experienced listeners were found to be significantly better than the scores of listeners with no musical experience also in this same error level range discrimination was significantly affected by the instrument tested for error levels of 16 and 24 discrimination scores were significantly but negatively correlated with measures of spectral incoherence and normalized centroid deviation on unaltered instrument spectra suggesting that the presence of dynamic spectral variations tends to increase the difficulty of detecting spectral alterations correlation between discrimination and a measure of spectral irregularity was comparatively low

on possible cues in profile analysis identification of the incremented component the ability to identify the frequency of the incremented component in a multitone complex was assessed in two conditions both using the same 11 tone complexes fixed frequency equilog spaced random phases in the standard plus signal complex all components had equal level except one of the inner nine which was incremented in level by an in phase addition of a sinusoid of the same frequency the signal the overall level and signal position were randomly varied on each presentation in the standard absent condition the standard plus signal complex was presented first and it was followed by a single sinusoid either at the signal frequency or at that of an adjacent component below or above in the standard present condition a complex with equal level components the standard complex was also presented on each trial either before or after the standard plus signal complex in random order these two complexes were followed by the single sinusoid in both conditions the subjects had to indica

relative weights for frequency glide detection using narrowband noise narrowband noise stimuli were used to derive relative weights for detecting frequency glides in a yes no procedure one stimulus set was restricted to the duration of the glide for the second stimulus set there were fringe noise bands preceding and following the glide for both sets the center frequency of the linear glide was either fixed at 1000hz or randomly chosen on each trial from the range of 800 1200hz relative weights as a function of time were derived using a linear model and the linear classification method a ahumada j vis 2 121 131 2002 sensitivity was better for the fixed than random frequency conditions and weight patterns from the random frequency conditions were less reliable than those obtained from the fixed frequency conditions the magnitudes of the relative weights tended to be larger for the second half of the stimulus suggesting that subjects paid more attention to the later than the earlier parts of stimuli in the random frequency conditions the linear model failed to account for subjects performance unless the stimuli were expressed in terms of relative changes in frequency rather than absolute frequency

explicit expressions of the generalized inverses and condensed cramer rules in this paper we obtain an explicit representation of the 2 inverse a 2 t s of a matrix a c m n with the prescribed range t and null space s as special cases new expressions for the moore penrose inverse a and drazin inverse a d are derived through explicit expressions we re derive the condensed cramer rules of werner for minimal norm least squares solution of linear equations ax b and propose two new condensed cramer rules for the unique solution of a class of singular system ax b x r a k b r a k k ind a finally condensed determinantal expressions for a a d a a a a a ndaa d are also presented

q state potts models in d dimensions migdal kadanoff approximation the first and second order phase transitions of the q state potts models are obtained in arbitrary dimension d critical and tricritical behaviours merge and annihilate at qc d clearing the way to first order transitions at q qc d by the condensation of effective vacancies the value of qc d decreases with increasing d from diverging as exp 2 d 1 at d to 1 to qc 2 3 81 cf exact value of 4 to lower values at d 2 for given d a changeover in critical behaviour occurs at q1 d as the critical fixed points merge from the potts lattice gas region to the undiluted potts limit it is suggested that the power law singularities of the percolation problem q to 1 have logarithmic corrections

the effect of onset asynchrony on relative weights in profile analysis decision weights were estimated in a profile analysis task to determine whether onset asynchronies between the signal component and the nonsignal components encourage the segregation of the signal relative to the other components the signal component onset was either synchronous or asynchronous with respect to the nonsignal components in the asynchronous conditions thresholds were higher and the decision weights were less efficient than in the synchronous conditions these data are largely consistent with a segregation hypothesis onset asynchrony encourages subjects to shift strategies from one of spectral shape discrimination toward one of intensity discrimination