anisotropic elastic plates with holes cracks inclusions subjected to out of plane bending moments the problems of anisotropic plates containing holes cracks and or inclusions have been studied extensively for two dimensional deformations although the correspondence between the two dimensional problems and the plate bending problems has been observed long time ago without clarifying the involved mathematical details one still cannot get the solutions of the plate bending problems directly from the solutions of the corresponding two dimensional problems based upon the correspondence relation recently we developed a stroh like formalism for the bending theory of anisotropic plates by this newly developed formalism most of the relations for bending problems can be organized into the forms for two dimensional problems thus by using the stroh like formalism the analytical solutions for problems of anisotropic plates with holes cracks inclusions subjected to out of plane bending moments can be obtained directly from the solutions of the corresponding two dimensional problems the stresses or resultant bending moments around the hole crack inclusion boundaries are also given explicitly in this paper since the explicit closed form solutions for the present cases have not been found in the literature comparison is made with some special cases of which the analytical solutions exist which shows that our solutions are exact simple and general

tight upper bounds on the minimum precision required of the divisor and the partial remainder in high radix division digit recurrence binary dividers are sped up via two complementary methods keeping the partial remainder in redundant form and selecting the quotient digits in a radix higher than 2 use of a redundant partial remainder replaces the standard addition in each cycle by a carry free addition thus making the cycles shorter deriving the quotient in high radix reduces the number of cycles by a factor of about h for radix 2 sup h to make the redundant partial remainder scheme work quotient digits must be chosen from a redundant set such as 2 2 in radix 4 the redundancy provides some tolerance to imprecision so that the quotient digits can be selected based on examining truncated versions of the partial remainder and divisor no closed form formula for the required precision in the partial remainder and divisor as a function of the quotient digit set and the range of the partial remainder is known we establish tight upper bounds on the required precision for the partial remainder and divisor the bounds are tight in the sense that each is only one bit over a well known simple lower bound we also discuss the implications of these bounds for the quotient digit selection process

sextic formalism in anisotropic elasticity for almost non semisimple matrix n abstract the sextic formalism of stroh for anisotropic elasticity leads to the eigen relation n p in which n is a 6 6 real matrix the orthogonality and closure relations as well as many other relations involving the eigenvalues p and the eigenvectors x are based on the assumption that n is simple or semisimple so that the six eigenvectors span a six dimensional space problems arise when n is non semisimple in fact there are problems even when n is almost non semisimple we present a modified formalism which is valid regardless of whether n is simple almost non semisimple or non semisimple the modified formalism does not apply when n is semisimple

on separability of bi hamiltonian chain with degenerated poisson structures separability of bi hamiltonian finite dimensional chains with two degenerated poisson tensors which have pfaffian quasi bi hamiltonian representation is proved

on the fly algorithms and sequential machines frougny has presented a method that generalizes various on the fly operations that have been presented mainly in connection with computer arithmetic first we shall trace the origin of this method to its source which is the celebrated paper of rabin and scott that introduced the notion of nondeterminism and the power set construction second we shall show that an understanding of this origin may lead to great quantitative improvements in applications of the method finally we shall show by a pathological example that the method as originally presented by frougny may result in circuits that are larger in terms of gates per step by two exponentiations than those that are constructed as described in the present paper

the differential cordic algorithm constant scale factor redundant implementation without correcting iterations the cordic algorithm is a well known iterative method for the efficient computation of vector rotations and trigonometric and hyperbolic functions basically cordic performs a vector rotation which is not a perfect rotation since the vector is also scaled by a constant factor this scaling has to be compensated for following the cordic iteration since cordic implementations using conventional number systems are relatively slow current research has focused on solutions employing redundant number systems which make a much faster implementation possible the problem with these methods is that either the scale factor becomes variable making additional operations necessary to compensate for the scaling or additional iterations are necessary compared to the original algorithm in contrast we developed transformations of the usual cordic algorithm which result in a constant scale factor redundant implementation without additional operations the resulting differential cordic algorithm dcordic makes use of on line most significant digit first redundant computation we derive parallel architectures for the radix 2 redundant number systems and present some implementation results based on logic synthesis of vhdl descriptions produced by a dcordic vhdl generator we finally prove that due to the lack of additional operations dcordic compares favorably with the previously known redundant methods in terms of latency and computational complexity

design of a radix 4 division unit with simple selection table a radix 4 division architecture is presented which partially overlaps the updating of the remainder with the digit selection procedure it is obtained by separating the radix 4 digit selection process into two concurrent substeps the proposed unit requires a simple selection table and involves a small extra expense for the additional hardware compared to the usual radix 4 division units four possible implementations are derived from the general model with different types of substeps the high level evaluation shows that the proposed architectures offer an efficient alternative

a vlsi algorithm for computing the euclidean norm of a 3d vector a digit recurrence algorithm for computing the euclidean norm of a three dimensional 3d vector which often appears in 3d computer graphics is proposed one of the three squarings required for the usual computation is removed and the other two squarings as well as the two additions are overlapped with the square rooting the euclidean norm is computed by iteration of carry propagation free additions shifts and multiplications by one digit different specific versions of the algorithm are possible depending on the radix the redundancy factor of the digit set and etc each version of the algorithm can be implemented as a sequential folded circuit or a combinational unfolded circuit which has a regular array structure suitable for vlsi

radix 2 division with over redundant quotient selection in this paper we present a new radix 2 division algorithm that uses a recurrence employing simple 3 to 2 digit carry free adders to perform carry free addition subtraction for computing the partial remainders in radix 2 signed digit form the quotient digit during any iteration of the division recursion is generated from the two most significant radix 2 digits of the partial remainder and independent of the divisor in over redundant radix 2 digit form i e with digits which belong to the digit set 2 1 0 1 2 the over redundant quotient digits are then converted to the conventional radix 2 digits belonging to the set 1 0 1 by using a reduction technique this division algorithm is well suited for ieee 754 standard operands belonging to the range 1 2 and is slightly faster than previously proposed radix 2 designs such as the radix 2 srt which do not employ input scaling since the quotient selection for such algorithms is a function of more than two most significant radix 2 digits of the partial remainder in comparison with the designs that employ input scaling the proposed design although slightly slower saves hardware required for scaling purposes

radix 16 signed digit division a two stage algorithm for fixed point radix 16 signed digit division is presented the algorithm uses two limited precision radix 4 quotient digit selection stages to produce the full radix 16 quotient digit the algorithm requires a two digit estimate of the initial partial remainder and a three digit estimate of the divisor to correctly select each successive quotient digit the normalization of redundant signed digit numbers requires accommodation of some fuzziness at one end of the range of numeric values that are considered normalized a set of general equations for determining the ranges of normalized signed digit numbers is derived another set of general equations for determining the precisions of estimates of the divisor and dividend are derived these two sets of equations permit design tradeoff analyses to be made with respect to the complexity of the model division the specific case of a two stage radix 16 signed digit division is presented the staged division algorithm used can be extended to other radices as long as the signed digital number representation used has certain properties

carry save multiplication schemes without final addition carry save multipliers require an adder at the last step to convert the carry sum representation of the most significant half of the result into a non redundant form this paper presents n spl times n multiplication schemes where this conversion is performed with a circuit operating in parallel with the carry save array the most relevant feature of the proposed multipliers is that the full 2n bit result is produced unlike similar multiplication schemes presented in the literature

higher radix square rooting a general discussion on nonrestoring square root algorithms is presented showing bounds and constraints delimiting the space of feasible algorithms for all the choices of radix digit set and representation of the partial remainder two classes of algorithms are then derived from the general discussion and it is shown how it is possible to determine two parameters with a relevant impact on the implementation the number of radicand bits to be inspected in order to obtain a starting value and the number of partial remainder bits to be examined for digit selection the algorithms for the specific case of radix 4 digit set 2 1 0 1 2 and partial remainder represented in carry save form are derived in order to show that the algorithms introduced can lead to better results than those obtained with algorithms previously presented

radix 4 square rot without initial pla a systematic derivation of a radix 4 square root algorithm using redundant residual and result is presented unlike other similar schemes it does not use a table lookup or pla for the initial step resulting in a simpler implementation without any time penalty the scheme can be integrated with division and incorporates an on the fly conversion and rounding of the result thus eliminating a carry propagate step to obtain the final result the result digit selection uses 3 bits of the result and 7 bits of the estimate of the residual

precision aware self quantizing hardware architectures for the discrete wavelet transform this paper presents designs for both bit parallel bp and digit serial ds precision optimized implementations of the discrete wavelet transform dwt with specific consideration given to the impact of depth the number of levels of dwt on the overall computational accuracy these methods thus allow customizing the precision of a multilevel dwt to a given error tolerance requirement and ensuring an energy minimal implementation which increases the applicability of dwt based algorithms such as jpeg 2000 to energy constrained platforms and environments additionally quantization of dwt coefficients to a specific target step size is performed as an inherent part of the dwt computation thereby eliminating the need to have a separate downstream quantization step in applications such as jpeg 2000 experimental measurements of design performance in terms of area speed and power for 90 nm complementary metal oxide semiconductor implementation are presented results indicate that while bp designs exhibit inherent speed advantages ds designs require significantly fewer hardware resources with increasing precision and dwt level a four level dwt with medium precision for example while the bp design is four times faster than the digital serial design occupies twice the area in addition to the bp and ds designs a novel flexible dwt processor is presented which supports run time configurable dwt parameters

over redundant digit sets and the design of digit by digit division units over redundant digit sets are defined as those ranging from spl minus s to s with s spl ges b b being the radix this paper presents new techniques for the direct computation of division that use an over redundant digit set for representing the quotient instead of simply redundant ones used previously in particular general criteria for synthesizing the digit selection rules and remainder updating are given for any radix and index of redundancy a methodology combining the use of over redundant digit sets with the prescaling of the divisor is also studied in order to achieve radix b division units with trivial digit selection functions it is also shown for the specific case of radix 4 that using a prescaling slightly wider than in a radix 4 unit by m d ercegovac and t lang 1990 possible to avoid the digit selection table the paper also presents a modified algorithm for on the fly conversion of the result into the irredundant form the proposed methodology can be considered as an alternative to existing division techniques

algorithm and architecture for logarithm exponential and powering computation an architecture for the computation of logarithm exponential and powering operations is presented in this paper based on a high radix composite algorithm for the computation of the powering function x sup y the algorithm consists of a sequence of overlapped operations 1 digit recurrence logarithm 2 left to right carry free lrcf multiplication and 3 online exponential a redundant number system is used and the selection in 1 and 3 is done by rounding except from the first iteration when selection by table look up is necessary to guarantee the convergence of the recurrences a sequential implementation of the algorithm with a control unit which allows the independent computation of logarithm and exponential is proposed and the execution times and hardware requirements are estimated for single and double precision floating point computations these estimates are obtained for radices from r 8 to r 1 024 according to an approximate model for the delay and area of the main logic blocks and help determining the radix values which lead to the most efficient implementations r 32 and r 128

presymplectic representation of bi hamiltonian chains liouville integrable systems which have bi hamiltonian representation of the gel fand zakharevich type are considered bi presymplectic representation of one casimir bi hamiltonian chains and weakly bi presymplectic representation of multi casimir bi hamiltonian chains are constructed the reduction procedure for poisson and presymplectic structures is presented

digit selection for srt division and square root the quotient digit selection in the srt division algorithm is based on a few most significant bits of the remainder and divisor where the remainder is usually represented in a redundant representation the number of leading bits needed depends on the quotient radix and digit set and is usually found by an extensive search to assure that the next quotient digit can be chosen as valid for all points remainder divisor in a set defined by the truncated remainder and divisor i e an uncertainty rectangle this work presents expressions for the number of bits needed from the truncated remainder and divisor the truncation parameters thus eliminating the need for a search through the truncation parameter space for validation the analysis is then extended to the digit selection in srt square root algorithms where it is shown that in general it may be necessary to increase the number of leading bits needed for digit determination in a combined divide and square root algorithm an easy condition to check the number of bits needed is established also checking the number of initial digits of the root may have to be found by other means e g by table look up the minimally redundant radix 4 combined divide and square root algorithm is finally analyzed and it is shown that in this case it can be implemented without such a special table to determine initial digits for the square root

leading guard digits in finite precision redundant representations redundant number representations are generally used to allow constant time additions based on the fact that only bounded carry ripples take place but carries may ripple out into positions which may not be needed to represent the final value of the result and thus a certain amount of leading guard digits are needed to correctly determine the result also when cancellation during subtractions occurs there may be nonzero digits in positions not needed to represent the result of the calculation it is shown here that for normal redundant digit sets with radix greater than two a single guard digit is sufficient to determine the value of such an arbitrary length prefix of leading nonzero digits this is also the case for the unsigned carry save representation whereas two guard digits are sufficient and may be necessary for additions in the binary signed digit and 2 s complement carry save representations thus only the guard digits need to be retained during sequences of additions and subtractions at suitable points the guard digits may then be converted into a single digit representing the complete prefix

two dimensional piezoelectricity part i eigensolutions of nondegenerate and degenerate materials abstract general solutions of two dimensional piezoelectricity which yield all solutions of 2 d boundary values problems are obtained by combining four complex conjugate pairs of independent eigensolutions each containing an arbitrary analytic function the forms of representation are fundamentally different for 14 different classes of nondegenerate and degenerate piezoelectric materials as determined by the multiplicity and types of eigenvalues degenerate materials possess high order eigensolutions in which the eigenvectors of equal and lower orders are intrinsically coupled such coupling is nonexistent in nondegenerate cases including the well known and analytically simple case with no multiple eigenvalues the present analysis is drastically simplified by using the compliance based formalism instead of the stiffness based extended eshelby stroh formalism explicit expressions are obtained for the eigensolutions the pseudometrics and the intrinsic tensors characterizing piezoelectric materials of every type

on the fly rounding computing arithmetic in implementations of operations based on digit recurrence algorithms such as division left to right multiplication and square root the result is obtained in digit serial form from most significant digit to least significant to reduce the complexity of the result digit selection and allow the use of redundant addition the result digit has values from a signed digit set as a consequence the result has to be converted to conventional representation which can be done on the fly as the digits are produced without the use of a carry propagate adder the authors describe three ways to modify this conversion process so that the result is rounded the resulting operation is fast because no carry propagate addition is needed the schemes described apply also to online arithmetic operations

a digit by digit algorithm for mth root extraction a general digit recurrence algorithm for the computation of the mth root with an m integer is presented in this paper based on the concept of completing the mth root a detailed analysis of the convergence conditions is performed and iteration independent digit selection rules are obtained for any radix and redundant digit set a radix 2 version for mth rooting is also studied together with closed formulas for both the digit selection rules and the number of bits required to perform correct selections

fast multiplication without carry propagate addition conventional schemes for fast multiplication accumulate the partial products in redundant form carry save or signed digit and convert the result to conventional representation in the last step this step requires a carry propagate adder which is comparatively slow and occupies a significant area of the chip in a vlsi implementation a report is presented on a multiplication scheme left to right carry free lrcf that does not require this carry propagate step the lrcf scheme performs the multiplication most significant bit first and produces a conventional sign and magnitude product most significant n bits by means of an on the fly conversion the resulting implementation is fast and regular and is very well suited for vlsi the lrcf scheme for general radix r and a radix 4 signed digit implementation are presented

a fast radix 4 division algorithm and its architecture in this paper we present a fast radix 4 division algorithm for floating point numbers this method is based on svoboda s division algorithm and the radix 4 redundant number system the algorithm involves a simple recurrence with carry free addition and employs prescaling of the operands in the proposed divider implementation each radix 4 digit belonging to set 3 3 of the quotient and partial remainder is encoded using two radix 2 digits belonging to the set 1 0 1 and this leads to hardware simplicity the quotient digits are determined by observing three most significant radix 2 digits of the partial remainder and independent of the divisor the architecture presented for the proposed algorithm is faster than previously proposed radix 4 dividers which require at least four digits of the partial remainder to be observed to determine quotient digits

deconstructing plane anisotropic elasticity part ii stroh s formalism sans frills abstract eigensolutions for all types of anisotropic elastic materials are obtained in terms of the eigenvalues and the anisotropic elastic stiffness the generalized eigenvectors and eigensolutions in the degenerate and extra degenerate cases are obtained by the derivative rule a complete set of unnormlized eigenvectors now given in terms of the elastic moduli define the barnett lothe tensors by the same expressions irrespective of material degeneracy explicit expressions of the barnett lothe tensors are obtained in various forms depending on the multiplicity of eigenvalues these expressions complement the alternative expressions of part i in terms of the elastic compliances a new family of extra degenerate materials is found suggesting the superabundance of such materials a concise proof of the equivalence of the eigensystems of the compliance based and elasticity based formalisms is given eigenrelations applicable to all cases of material degeneracy are presented in both three dimensional and six dimensional matrix formalisms

a remark on reducing iteration time when result digit is zero for radix 2 srt division and square root with redundant remainders in a previous paper by p montuschi and l ciminiera ibid vol 42 no 2 p239 246 feb 1993 an architecture for shared radix 2 division and square root has been presented whose main characteristic is the ability to avoid any addition subtraction when the digit 0 has been selected here we emphasize the characteristics of the digit selection mechanism used by montuschi and ciminiera by presenting a small modification of the digit selection hardware which has the benefit to further reduce the computation delay with respect to the time estimated in that work

quasi lagrangian systems of newton equations systems of newton equations of the form q 12a 1 q k with an integral of motion quadratic in velocities are studied these equations generalize the potential case when a i the identity matrix and they admit a curious quasi lagrangian formulation which differs from the standard lagrange equations by the plus sign between terms a theory of such quasi lagrangian newton qln systems having two functionally independent integrals of motion is developed with focus on two dimensional systems such systems admit a bi hamiltonian formulation and are proved to be completely integrable by embedding into five dimensional integrable systems they are characterized by a linear second order partial differential equation pde which we call the fundamental equation fundamental equations are classified through linear pencils of matrices associated with qln systems the theory is illustrated by two classes of systems separable potential systems and driven systems new separation variables for driven systems are foun

redundant and on line cordic application to matrix triangularization and svd several modifications to the cordic method of computing angles and performing rotations are presented 1 the use of redundant carry free addition instead of a conventional carry propagate one 2 a representation of angles in a decomposed form to reduce area and communication bandwidth 3 the use of on line addition left to right digit serial addition to replace shifters by delays and 4 the use of online multiplication square root and division to compute scaling factors and perform the scaling operations the modifications improve the speed and the area of cordic implementations the proposed scheme uses efficiently floating point representations the application of the modified cordic method to matrix triangularization by givens rotations and to the computation of the singular value decomposition svd are discussed

new svoboda tung division the paper presents a general theory for developing new svoboda tung or simply nst division algorithms not suffering the drawbacks of the classical svoboda tung or simply st method nst avoids the drawbacks of st by proper recoding of the two most significant digits of the residual before selecting the most significant digit of this recoded residual as the quotient digit nst relies on the divisor being in the range 1 1 spl delta where spl delta is a positive fraction depending upon 1 the radix 2 the signed digit set used to represent the residual and 3 the recoding conditions of the two most significant digits of the residual if the operands belong to the ieee std range 1 2 they have to be conveniently prescaled in that case nst produces the correct quotient but the final residual is scaled by the same factor as the operands therefore nst is not useful in applications where the unsealed residual is necessary an analysis of nst shows that previously published algorithms can be derived from the general theory proposed in the paper moreover nst reveals a spectrum of new possibilities for the design of alternative division units for a given radix b the number of different algorithms of this kind is b sup 2 4

intrinsic characterization of the variable separation in the hamilton jacobi equation the nonorthogonal separation of variables in the hamilton jacobi equation corresponding to a natural hamiltonian h 12gijpipj v with a metric tensor of any signature is intrinsically characterized by geometrical objects on the riemannian configuration manifold killing vectors killing tensors and killing webs comparisons with previous characterizations and some illustrative examples are given

higher radix square root with prescaling a scheme for performing higher radix square root based on prescaling of the radicand is presented to reduce the complexity of the result digit selection the scheme requires several steps namely multiplication for prescaling the radicand square root multiplication for prescaling for the division and division online algorithms are used to reduce the overall time and pipelining to reuse the different modules an estimate of the execution time for a radix 256 unit for double precision square root and a comparison with other implementations indicate that the proposed approach is an alternative to consider when designing a square root unit

theory of separability of multi hamiltonian chains the theory of separability of one casimir bi hamiltonian chains is extended onto unsplit multi casimir bi hamiltonian chains multi casimir extensions of the known one casimir chains are constructed

a class of poisson nijenhuis structures on a tangent bundle equipping the tangent bundle tq of a manifold with a symplectic form coming from a regular lagrangian l we explore how to obtain a poisson nijenhuis structure from a given type 1 1 tensor field j on q it is argued that the complete lift jc of j is not the natural candidate for a nijenhuis tensor on tq but plays a crucial role in the construction of a different tensor r which appears to be the pullback under the legendre transform of the lift of j to t q we show how this tangent bundle view brings new insights and is capable also of producing all important results which are known from previous studies on the cotangent bundle in the case when q is equipped with a riemannian metric the present approach further paves the way for future generalizations

a modified partial product generator for redundant binary multipliers due to its high modularity and carry free addition a redundant binary rb representation can be used when designing high performance multipliers the conventional rb multiplier requires an additional rb partial product rbpp row because an error correcting word ecw is generated by both the radix 4 modified booth encoding mbe and the rb encoding this incurs in an additional rbpp accumulation stage for the mbe multiplier in this paper a new rb modified partial product generator rbmppg is proposed it removes the extra ecw and hence it saves one rbpp accumulation stage therefore the proposed rbmppg generates fewer partial product rows than a conventional rb mbe multiplier simulation results show that the proposed rbmppg based designs significantly improve the area and power consumption when the word length of each operand in the multiplier is at least 32 bits these reductions over previous nb multiplier designs incur in a modest delay increase approximately 5 percent the power delay product can be reduced by up to 59 percent using the proposed rb multipliers when compared with existing rb multipliers

quasi bi hamiltonian systems and separability two quasi bi hamiltonian systems with three and four degrees of freedom are presented these systems are shown to be separable in terms of nijenhuis coordinates moreover the most general pfaffian quasi bi hamiltonian system with an arbitrary number of degrees of freedom is constructed in terms of nijenhuis coordinates and its separability is proved

two dimensional piezoelectricity part ii general solution green s function and interface cracks abstract the complete solution space of a piezoelectric material is the direct sum of several orthogonal eigenspaces one for each distinct eigenvalue each one of the 14 different classes of piezoelectric materials has a distinct form of the general solution expressed in terms of the eigenvectors of the zeroth and higher orders and a kernel matrix containing analytic functions when these functions are chosen to be logarithmic one obtains in a unified way green s function of the infinite space as a single 8 8 matrix function g for the various load cases of concentrated line forces dislocations and a line charge this expression of green s function is valid for all classes of nondegenerate and degenerate materials with an appropriate choice of the parameters it reduces to the solution of a half space with concentrated line forces at a boundary point and with dislocations in the displacements as another application eigenvalues and eigensolutions are obtained for the bimaterial interface crack problem

on the fly conversion of redundant into conventional representations an algorithm to convert redundant number representations into conventional representations is presented the algorithm is performed concurrently with the digit by digit generation of redundant forms by schemes such as srt division it has a step delay roughly equivalent to the delay of a carry save adder and simple implementation the conversion scheme is applicable in arithmetic algorithms such as nonrestoring division square root and on line operations in which redundantly represented results are generated in a digit by digit manner from most significant to least significant

choices of operand truncation in the srt division algorithm the paper presents an analysis of the number of partial remainder digits and divisor bits that must be examined in the srt division algorithm the number of examined digits is found to be the same for both signed digit and 2s complement partial remainder representations and appears to increase as 3log sub 2 r approximately where r is the radix of the divider in some cases it proves advantageous to examine a fractional number of remainder digits by inspecting more positive than negative bits

computation and formal verification of srt quotient and square root digit selection tables we present a comprehensive self contained and mechanically verified proof of correctness of a maximally redundant srt design for floating point division and square root extraction supported by verified procedures that 1 test the admissibility of a proposed digit selection table 2 determine the minimal dimensions of an admissible table for a given arbitrary radix and 3 generate these tables for square root extraction we also provide a verified procedure for generating an initial approximation that meets the accuracy requirement of the algorithm and ensures that the digit selection index derived from successive partial roots remains static throughout the computation a radix 8 instantiation of these algorithms has been implemented in the floating point unit of the amd processor code named steamroller to ensure their correctness all of our results and procedures have been formalized and mechanically checked by the acl2 prover we present evidence of the value of this approach by comparing it to that of a more conventional published paper that reports similar results which are shown to be fatally flawed

separability preserving dirac reductions of poisson pencils on riemannian manifolds dirac deformation of poisson operators of arbitrary rank is considered the question when dirac reduction does not destroy linear poisson pencils is studied a class of separability preserving dirac reductions in the corresponding quasi bi hamiltonian systems of benenti type is discussed two examples of such reductions are given

constant factor redundant cordic for angle calculation and rotation a constant factor redundant cordic cfr cordic scheme where the scale factor is kept constant while an angle for plane rotations is computed is developed the direction of rotation is determined from an estimate of the sign and convergence is assured by suitably placed correcting iterations the number of iterations in the cordic rotation unit is reduced by about 25 by expressing the direction of the rotation in radix 2 and radix 4 and conversion to conventional representation is done on the fly the performance of cfr cordic is estimated and compared with that of previously proposed schemes it is found to provide an execution time similar to that of redundant cordic with a variable scaling factor with a significant saving in area

high radix multiplier dividers theory design and hardware this paper describes the theory and design of digital high radix multiplier dividers patent pending the theory of high radix division is extended to high radix multiplier dividers that can perform fused multiplication and division operations using a single recurrence relation with the fused implementation of multiplication and division the two operations can be executed using a single instruction implying only a single rounding operation the recurrence relation is described the quotient digit selection function derived and important design parameters together with their optimal values and relations are defined efficient design procedure and implementation hardware are described and important system parameter values for various radix systems computed compared to pure dividers the multiplier divider requires a slightly more complex data path and quotient digit selection function

simple radix 4 division with operands scaling a radix 4 division algorithm with operands scaling is proposed the algorithm uses a recurrence with redundant addition carry save or signed digit and combines simple scaling with a quotient selection function that depends only on the estimate of the partial remainder and is independent of the divisor the scheme results in a significant speedup with respect to both the radix 2 and radix 4 without scaling

a modified lekhnitskii formalism a la stroh for anisotropic elasticity and classifications of the 6 6 matrix n the stroh formalism for two dimensional deformations of anisotropic elastic materials computes the eigenvalue p and the eigenvector a from an eigenrelation the vector b is then determined from a depending on the number of repeated eigenvalues p and the number of independent eigenvectors a b the system has it can be classified into six groups the lekhnitskii formalism has no eigenrelation to speak of we present a modified lekhnitskii formalism that computes the eigenvalue p and the eigenvector b from an eigenrelation the vector a is then determined from b thus the modified lekhnitskii formalism is a dual to the stroh formalism not only does the modified formalism enable us to do the classifications it is much simpler than using the stroh formalism the six groups are the sp ss d1 d2 ed and es groups the es group does not exist for a real material the ss group that has p 1 p 2 p 3 and the d2 group that has p 1 p 2 p 3 can be identified without computing the eigenvalues p and the eigenvectors a b we show that the repeated eigenvalue p 1 p 2 in the ss and d2 groups is simply a root of the quadratic equation l 2 0 we present an explicit expression of p 3 for the ss group the ed group that has three identical p can also be identified without computing p and a b however we do present an explicit expression of p we show that monoclinic materials with the symmetry plane at x 3 0 cannot belong to the ed group the identification of the sp and d1 groups is the only one that requires computation of p but not a b for special classes of materials however they can be identified without computing p in all cases the eigenvectors and the generalized eigenvectors are obtained explicitly

radix 2 division algorithms with an over redundant digit set this paper presents a derivation of four radix 2 division algorithms by digit recurrence each division algorithm selects a quotient digit from the over redundant digit set 2 1 0 1 2 and the selection of each quotient digit depends only on the two most significant digits of the partial remainder in a redundant representation two algorithms use a two s complement representation for the partial remainder and carry save additions and the other two algorithms use a binary signed digit representation for the partial remainder and carry free additions three algorithms are novel the fourth algorithm has been presented before results from the synthesized netlists show that two of our fastest algorithms achieve an improvement of 10 percent in latency per iteration over a standard radix 2 srt algorithm at the cost of 36 percent more power and 50 percent more area

digit set conversions generalizations and applications the problem of digit set conversion for fixed radix is investigated for the case of converting into a non redundant as well as into a redundant digit set conversion may be from very general digit sets and covers as special cases multiplier recodings additions and certain multiplications we generalize known algorithms for conversions into non redundant digit sets as well as apply conversion to generalize the o log n time algorithm for conditional sum addition using parallel prefix computation and a comparison is made with standard carry lookahead techniques examples on multi operand addition are used to illustrate the generality of this approach o 1 time algorithms for converting into redundant digit sets are generalized based on a very simple lemma which provides a framework for all conversions into redundant digit sets applications in multiplier recoding and partial product accumulation are used as exemplifications

a high speed reduced size adder under left to right input arrival an efficient parallel adder under left to right input arrival is proposed making full use of the delay of the input arrival it produces the sum within a small constant delay after the arrival of the final bits its amount of hardware is proportional to the operand length it can be applied to the quotient conversion in an array divider

two families of nonstandard poisson structures for newton equations two families of nonstandard two dimensional poisson structures for systems of newton equations are studied they are closely related either with separable systems or with the so called quasi lagrangian systems a theorem characterizing the general form of bi hamiltonian formulation for separable systems in two and in n dimensions is formulated and proved

separation of variables in quasi potential systems of bi cofactor form we perform variable separation in the quasi potential systems of equations of the form q a 1 k a 1 k w herea and a are killing tensors by embedding these systems into a bi hamiltonian chain and by calculating the corresponding darboux nijenhuis coordinates on the symplectic leaves of one of the hamiltonian structures of the system we also present examples of the corresponding separation coordinates in two and three dimensions

separable systems with quadratic in momenta first integrals the separability theory of hamiltonian systems on riemannian manifolds is reviewed and developed particular attention is paid to the systems generated by the so called special conformal killing tensors i e benenti systems then infinitely many new classes of separable systems are constructed by appropriate deformations of benenti class systems

a state space formalism for anisotropic elasticity part i rectilinear anisotropy a state space formalism for anisotropic elasticity including the thermal effect is developed a salient feature of the formalism is that it bridges the compliance based and stiffness based formalisms in a natural way the displacement and stress components and the thermoelastic constants of a general anisotropic elastic material appear explicitly in the formulation yet it is simple and clear this is achieved by using the matrix notation to express the basic equations and grouping the stress in such a way that it enables us to cast neatly the three dimensional equations of anisotropic elasticity into a compact state equation and an output equation the homogeneous solution to the state equation for the generalized plane problem leads naturally to the eigen relation and the sextic equation of stroh extension twisting bending temperature change and body forces are accounted for through the particular solution based on the formalism the general solution for generalized plane strain and generalized torsion of an anisotropic elastic body are determined in an elegant manner

bi hamiltonian representation of stackel systems it is shown that linear separation relations are fundamental objects for integration by quadratures of stackel separable liouville integrable systems the so called stackel systems these relations are further employed for the classification of stackel systems moreover we prove that any stackel separable liouville integrable system can be lifted to a bi hamiltonian system of gel fand zakharevich type in conjunction with other known result this implies that the existence of bi hamiltonian representation of liouville integrable systems is a necessary condition for stackel separability

painleve conjecture revisited the discovery of new integrable two dimensional hamiltonian systems is reported the analytic structure of the solutions makes necessary the generalization of the painlev e conjecture a widely used integrability criterion such a generalization is presented which the authors believe should replace the usual conjecture for two dimensional hamiltonian systems it is indeed compatible with all the systems already found and in addition leads to still new integrable cases

driven newton equations and separable time dependent potentials we present a class of time dependent potentials in rn that can be integrated by separation of variables by embedding them into so called cofactor pair systems of higher dimension we are led to a

a class of nonconservative lagrangian systems on riemannian manifolds we generalize results of rauch wojciechowski marciniak and lundmark concerning a class of nonconservative lagrangian systems from the euclidean to the riemannian case

digit serial methods with applications to division and square root we present a generic digit serial method dsm to compute the digits of a real number v bounds on these digits and on the errors in the associated estimates of v formed from these digits are derived to illustrate our results we derive such bounds for a parameterized family of high radix algorithms for division and square root these bounds enable a dsm designer to determine for example whether a given choice of parameters allows rapid formation and rounding of its approximation to v