from kontsevich witten to linear hodge integrals via virasoro operators we give a proof of alexandrov s conjecture on a formula connecting the kontsevich witten and hodge tau functions using only the virasoro operators this formula has been confirmed up to an unknown constant factor in this paper we show that this factor is indeed equal to one by investigating series expansions for the lambert w function on different points we give a proof of alexandrov s conjecture on a formula connecting the kontsevich witten and hodge tau functions using only the virasoro operators this formula has been confirmed up to an unknown constant factor in this paper we show that this factor is indeed equal to one by investigating series expansions for the lambert w function on different points

separate joint optimization of error feedback and coordinate transformation for roundoff noise minimization in two dimensional state space digital filters this paper is concerned with the minimization of roundoff noise subject to l sub 2 norm dynamic range scaling constraints in two dimensional 2 d state space digital filters two methods are proposed with the first one using error feedback alone and the second one using joint error feedback and coordinate transformation optimization in the first method several techniques for the determination of optimal full scale block diagonal diagonal and scalar error feedback matrices for a given 2 d state space digital filter are proposed in the second method an iterative approach for minimizing the roundoff noise under l sub 2 norm dynamic range scaling constraints is developed by jointly optimizing a scalar error feedback matrix and a coordinate transformation matrix which may be regarded as an alternative approach to the conventional method for synthesizing the optimal 2 d filter structure with minimum roundoff noise an analytical method for the joint optimization of a general error feedback matrix and a coordinate transformation matrix under the scaling constraints is also proposed a numerical example is presented to illustrate the utility of the proposed techniques

a levinson type algorithm for modeling fast sampled data the standard discrete time autoregressive model is poorly suited for modeling series obtained by sampling continuous time processes at fairly rapid rates large computational errors can occur when the levinson algorithm is used to estimate the parameters of this model because the toeplitz covariance matrix is ill suited for inversion an alternative model is developed based on an incremental difference operator rather than the shift operator it is shown that as the sampling period goes to zero unlike the standard autoregressive parameters the coefficients of this model converge to certain parameters that depend directly on the statistics of the continuous time process a levinson type algorithm for efficiently estimating the parameters of this model is derived numerical examples are given to show that when the sampling interval is small this algorithm is considerably less sensitivity to arithmetic roundoff errors than the levinson algorithm

kp hierarchy for hodge integrals abstract starting from the elsv formula we derive a number of new equations on the generating functions for hodge integrals over the moduli space of complex curves this gives a new simple and uniform treatment of certain known results on hodge integrals like witten s conjecture virasoro constrains faber s g conjecture etc among other results we show that a properly arranged generating function for hodge integrals satisfies the equations of the kp hierarchy

noise reduction strategies for digital filters error spectrum shaping versus the optimal linear state space formulation the noise reduction performance of error spectrum shaping ess structures and of the optimal linear state space lss structure are compared for second order digital filter sections it is shown that optimal direct form 1 and direct form 2 ess realizations have a higher signal to noise ratio than the optimal lss structure in practice suboptimal ess structures with simple hardware implementations are of greater interest several of these implementations are considered and optimal values for the ess coefficients in these structures are derived for filters with zeros at z 1 it is shown that some of the simple ess structures can outperform the optimal lss structure for elliptic filters it is shown that several of the suboptimal ess structures perform poorly but that others still perform well

a new scheme for designing iir filters with finite wordlength coefficients a scheme for designing iir digital filters with specified coefficient wordlengths based on successive digitization and reoptimization is presented and compared to related approaches via a number of design examples the ordered digitization of the coefficients based on their sensitivity with each digitization followed by reoptimization of the remaining coefficients results in a much reduced computation effort in comparison to related design approaches

on the cauchy index of linear systems the cauchy index for linear siso systems is given by the signature of the cross gramian matrix w co defined by fernando and nicholson 1 this index is useful in the qualitative understanding of systems and the importance of this index in internally balanced model reduction is illustrated

enumerative geometry tau functions and heisenberg virasoro algebra in this paper we establish relations between three enumerative geometry tau functions namely the kontsevich witten hurwitz and hodge tau functions the relations allow us to describe the tau functions in terms of matrix integrals virasoro constraints and kac schwarz operators all constructed operators belong to the algebra or group of symmetries of the kp hierarchy

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

comparative study of finite wordlength effects in shift and delta operator parameterizations the sensitivity of the transfer function to finite wordlength effect errors in the implementation of the coefficients of both shift operator and delta operator parameterizations is analyzed the relationships between optimal realization sets in shift and delta operator are established it is shown that the optimal realization in delta operator parameterizations have better performance than those in their shift operator counterparts when the poles of the systems are clustered around z 1 a numerical example is given

jointly optimal high order error feedback and realization for roundoff noise minimization in 1 d and 2 d state space digital filters joint optimization of high order error feedback and state space realization for minimizing roundoff noise at filter output subject to l2 scaling constraints is investigated for one and two dimensional state space digital filters linear algebraic techniques that convert the problems at hand into an unconstrained optimization problem are explored and an efficient quasi newton algorithm is then applied to solve the unconstrained optimization problem iteratively in this connection closed form formulas are derived for fast and accurate gradient evaluation finally case studies are presented to demonstrate that the high order error feedback does offer much improved performance and that the proposed joint optimization is superior relative to a sequentially optimized system where the state space coordinate transformation and high order error feedback matrices are optimized separately

design of multiplierless elliptic iir filters with a small quantization error we present a new technique for the design of multiplierless iir elliptic filters the multiplierless filter has all multiplication constants implemented with a small number of shifters and adders the proposed technique is based on sensitivity analysis an analytical expression for amplitude response sensitivity is derived for the filter structures consisting of two allpass subfilters in parallel it is shown that the amplitude response sensitivity to some constant x can be expressed as a product of the filter reflectance function and the phase sensitivity of the allpass section that implements the constant the closed form expressions for the phase sensitivities of the first and second order allpass sections are also developed it is shown in the paper that the n 1 2 most sensitive constants can be directly controlled by the transfer function parameters if the transfer function is derived by the bilinear transformation from an elliptic minimal q factors emqf analog prototype this way n 1 2 multiplication constants can be implemented without quantization leaving the filter characteristic strictly elliptic this is achieved for a class of low noise allpass sections and for the wave lattice digital filter as well the quantization of the remaining n 1 2 less sensitive constants is performed using the phase tolerance scheme and phase sensitivity functions the proposed design technique is straight forward and consequently very fast the application is demonstrated on the examples of narrowband wideband and halfband filters

alternative discrete time operators an algorithm for optimal selection of parameters we consider the issue of parameter sensitivity in models based on alternative discrete time operators adtos a generic first order adto is proposed that encompasses all the known first order adtos introduced so far in the literature new bounds on the operator parameters are derived and a new algorithm is given for optimally selecting the parameters to give minimum parameter sensitivity

connecting the kontsevich witten and hodge tau functions by the widehat gl infty operators in this paper we present an explicit formula that connects the kontsevich witten tau function and the hodge tau function by differential operators belonging to the widehat gl infty group indeed we show that the two tau functions can be connected using virasoro operators this proves a conjecture posted by alexandrov in from hurwitz numbers to kontsevich witten tau function a connection by virasoro operators letters in mathematical physics doi 10 1007 s11005 013 0655 0 2014

noise reduction in recursive digital filters using high order error feedback the problem of solving the optimal minimum noise error feedback coefficients for recursive digital filters is addressed in the general high order case it is shown that when minimum noise variance at the filter output is required the optimization problem leads to set of familiar wiener hopf or yule walker equations demonstrating that the optimal error feedback can be interpreted as a special case of wiener filtering as an alternative to the optimal solution the formulas for suboptimal error feedback with symmetric or antisymmetric coefficients are derived in addition the design of error feedback using power of two coefficients is discussed the efficiency of high order error feedback is examined by test implementations of the set of standard filters it is concluded that error feedback is a very powerful and versatile method for cutting down the quantization noise in any classical infinite impulse response iir filter implemented as a cascade of second order direct form sections the high order schemes are attractive for use with high order direct form sections

improved bounds for some nonstandard problems in generalized heat conduction abstract we consider the maxwell cattaneo system of equations for generalized heat conduction where the temperature and heat flux satisfy a nonstandard auxiliary condition which prescribes a combination of their values initially and at a later time we obtain l 2 bounds for the temperature and heat flux by means of lagrange identities these bounds extend the range of validity for the parameter in the nonstandard condition under a constraint on the coefficients in the differential equations

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

roundoff noise minimization using delta operator realizations digital filters the authors examine the advantages of using delta operator state space realizations rather than shift operator realizations of transfer functions in terms of minimizing roundoff noise gain they give several conditions under which the optimal roundoff noise gain for delta operator realizations is smaller than the optimal gain for shift operator realizations they then illustrate that even sparse and hence nonoptimal delta operator realizations can have smaller roundoff noise gain than the optimal shift operator realizations

error feedback in a class of orthogonal polynomial digital filter structures a method of applying error feedback in three digital ladder and lattice filter structures derived from the theory of orthogonal polynomials is proposed it is shown that error feedback leads to significant improvement in both the roundoff noise characteristics and limit cycle behavior in particular the error compensated one multiplier structure is superior to the uncompensated normalized ladder this results in significant savings in both hardware and speed of implementation

roundoff noise minimization and pole zero sensitivity in fixed point digital filters using residue feedback the problem of designing a finite wordlength fixed point realization of an nth order digital filter which uses residue feedback to minimize the output roundoff noise subject to l 2 scaling is considered the new structures require n extra additions but no more multiplications than the earlier low noise structures of mullis and roberts and have lower roundoff noise for sufficiently narrow bandwidth filters a new set of filter invariants called the residue modes are defined which characterize the new low noise structures and determine the output noise variance if the sum of the residue modes is less than the sum of the second order modes of mullis and roberts then lower roundoff noise is achieved every filter structure is shown to define a unique symmetrizing matrix q pole zero sensitivities and a new noise measure are defined in terms of q numerical results are included which compare the noise and pole zero sensitivity characteristics of different filter structures

optimal finite wordlength linear quadratic regulation the authors consider the design of an lqg regulator implemented in fixed point arithmetic which is optimized with respect to its state wordlength and structure the approach extends previous design methods so that arithmetic errors are directly included the optimal finite controller gains are computed as a function of the arithmetic wordlength subject to an l sub 2 scaling constant on the controller states the design method used permits the cost for the near optimal structure to be developed so that this can then be used as a basis for comparison to other less complex structures

qualitative aspects in dual phase lag thermoelasticity we consider the system of dual phase lag thermoelasticity proposed by chandrasekharaiah and tzou first we prove that the solutions of the problem are generated by a semigroup of quasi contractions thus the problem of the third order in time is well posed then the exponential stability is investigated finally the spatial behavior of solutions is analyzed in a semi infinite cylinder and a result on the domain of influence is obtained

low roundoff noise augmented realizations of iir filters the scaling factor in a direct form ii infinite impulse response iir digital filter depends on only the system poles not its zeros adding pole zero cancellation pairs changes the scaling factor and the associated roundoff noise without affecting the transfer function it is shown that with l sub 2 scaling the upper bound of the noise reduction for direct form ii iir digital filters using k pole zero cancellation pairs is about 6k db the authors consider parallel form 3p to be the basic structure in which each subfilter is augmented with one or two pole zero cancellation pairs and realized in direct form ii they seek an iterative optimization process for each subfilter to find the optimal pole zero cancellation pairs that minimize the roundoff noise experimental results indicate the global optimal solution for these augmented subfilters can be obtained by using simple criteria to determine the starting points of the iterative processes alternative structures illustrative examples and overflow limit cycles are examined

delay replacement in direct form structures a class of high order low sensitivity digital filter structures is generated by means of a direct transformation of a high order direct form df structure the proposed structures have low coefficient dynamic range and exhibit low coefficient frequency response sensitivity and roundoff noise characteristics yet require only 3n multiplications per output sample as compared to 2n in the direct form case

on two classes of nonstandard parabolic problems abstract in this paper the authors continue an investigation of nonstandard parabolic problems this investigation was motivated by one of the suggested methods for stabilizing ill posed problems for evolution equations two different methods are employed methods which in a sense complement one another

roundoff noise minimization for 2 d state space digital filters using joint optimization of error feedback and realization the joint optimization problem of error feedback and realization for two dimensional 2 d state space digital filters to minimize the effects of roundoff noise at the filter output subject to l2 norm dynamic range scaling constraints is investigated it is shown that the problem can be converted into an unconstrained optimization problem by using linear algebraic techniques the unconstrained optimization problem at hand is then solved iteratively by applying an efficient quasi newton algorithm with closed form formulas for key gradient evaluation analytical details are given as to how the proposed technique can be applied to the cases where the error feedback matrix is a general block diagonal diagonal or block scalar matrix a case study is presented to illustrate the utility of the proposed technique