blocks of small defect abstract let g be a finite solvable group let p be a prime such that p 5 and o p g 1 and we denote g p p n then g contains a block of defect less than or equal to 3 n 5 let g be a finite solvable group and let p a be the largest power of p dividing 1 for an irreducible character of g we show that g f g p p 3 a for p 5

a modified split radix fft with fewer arithmetic operations recent results by van buskirk have broken the record set by yavne in 1968 for the lowest exact count of real additions and multiplications to compute a power of two discrete fourier transform dft here we present a simple recursive modification of the split radix algorithm that computes the dft with asymptotically about 6 fewer operations than yavne matching the count achieved by van buskirk s program generation framework we also discuss the application of our algorithm to real data and real symmetric discrete cosine transforms where we are again able to achieve lower arithmetic counts than previously published algorithms

recursive generation of the dif fft algorithm for 1 d dft it is shown by using the walsh hadamard matrix and a conversion matrix a dif fft algorithm can be developed for the radix 2 discrete fourier transform dft the recursive generation of the dft matrix leads to its sparse matrix factorization the flowgraph based on these sparse matrices is identical to that of the dft fft algorithm

on hardware for computing exponential and trigonometric functions this paper presents new fast hardware for computing the exponential function sine and cosine the main new idea is to use low precision arithmetic components to approximate high precision computations and then to correct very quickly the approximation error periodically so that the effect is to get high precision computation at near low precision speed the algorithm used in the paper is a nontrivial modification of the well known cordic algorithm and might be applicable to the computation of other functions than the ones presented

brauer characters and normal sylow p subgroups abstract in this paper we study some variations of the well known ito michler theorem for p brauer characters using various inequalities involving the p brauer character degrees of finite groups several new criteria for the existence of a normal sylow p subgroup of finite groups are obtained using the p parts and p parts of the p brauer character degrees

input and or output pruning of composite length ffts using a dif dit transform decomposition pruned fast fourier transforms ffts can be efficient alternatives to compute dfts when the input vector is zero padded and or several output elements are not required in this correspondence a new method to prune composite length ffts is proposed the proposed pruning method uses decimation in frequency dif and decimation in time dit to decompose a dft into stages of smaller dfts the pruning process is carried out on the input stage and the output stage of the decomposed transform the proposed pruning method is flexible since it can perform input and or output pruning over any composite length fft action that no other pruning method reported in the literature can carry out additionally no restriction exists with the number of consecutive inputs and consecutive outputs that can be used finally it is shown that the proposed pruning method generates efficient pruned power of three and power of two length ffts

on computing the split radix fft this paper presents an efficient fortran program that computes the duhamel hollmann split radix fft an indexing scheme is used that gives a three loop structure for the split radix fft that is very similar to the conventional cooley tukey fft both a decimation in frequency and a decimation in time program are presented an arithmetic analysis is made to compare the operation count of the cooley tukey fft fo several different radixes to that of the split radix fft the split radix fft seems to require the least total arithmetic of any power of two dft algorithm

radix k fft s using k point convolutions a new class of dft algorithms so called c ffts is presented the radix k c ffts are derived from structures composed of k point convolutions and k point dfts radix 3 4 and 6 c ffts have the smallest arithmetical complexities and in contrast to other existing algorithms this fact does not depend on the complex number base used additionally the transformation of c ffts into algorithms for real valued data is straightforward the analysis of radix 6 c ffts leads to a refinement of the split radix fft concept the refinement can be used for deriving improved radix k ffts when k is a product of mutually prime numbers

the existence of current vortex sheets in ideal compressible magnetohydrodynamics we prove the local in time existence of solutions with a surface of current vortex sheet tangential discontinuity of the equations of ideal compressible magnetohydrodynamics in three space dimensions provided that a stability condition is satisfied at each point of the initial discontinuity this paper is a natural completion of our previous analysis trakhinin in arch ration mech anal 177 331 366 2005 where a sufficient condition for the weak stability of planar current vortex sheets was found and a basic a priori estimate was proved for the linearized variable coefficients problem for nonplanar discontinuities the original nonlinear problem is a free boundary hyperbolic problem since the free boundary is characteristic the functional setting is provided by the anisotropic weighted sobolev spaces h m the fact that the kreiss lopatinski condition is satisfied only in a weak sense yields losses of derivatives in a priori estimates therefore we prove our existence theorem by a suitable nash moser type iteration scheme

a priori estimates for 3d incompressible current vortex sheets we consider the free boundary problem for current vortex sheets in ideal incompressible magneto hydrodynamics it is known that current vortex sheets may be at most weakly neutrally stable due to the existence of surface waves solutions to the linearized equations the existence of such waves may yield a loss of derivatives in the energy estimate of the solution with respect to the source terms however under a suitable stability condition satisfied at each point of the initial discontinuity and a flatness condition on the initial front we prove an a priori estimate in sobolev spaces for smooth solutions with no loss of derivatives the result of this paper gives some hope for proving the local existence of smooth current vortex sheets without resorting to a nash moser iteration such result would be a rigorous confirmation of the stabilizing effect of the magnetic field on kelvin helmholtz instabilities which is well known in astrophysics

characters of relative p degree over normal subgroups let z be a normal subgroup of a finite group g let irr z be an irreducible complex character of z and let p be a prime number if p does not divide the integers 1 1 for all irr g lying over then we prove that the sylow p subgroups of g z are abelian this theorem which generalizes the gluck wolf theorem to arbitrary finite groups is one of the principal obstacles to proving the celebrated brauer height zero conjecture

defect zero p blocks for finite simple groups we classify those finite simple groups whose brauer graph or decomposition matrix has a p block with defect 0 completing an investigation of many authors the only finite simple groups whose defect zero p blocks remained unclassified were the alternating groups an here we show that these all have a p block with defect 0 for every prime p 5 this follows from proving the same result for every symmetric group sn which in turn follows as a consequence of the t core partition conjecture that every non negative integer possesses at least one t core partition for any t 4 for t 17 we reduce this problem to lagrange s theorem that every non negative integer can be written as the sum of four squares the only case with t 17 that was not covered in previous work was the case t 13 this we prove with a very different argument by interpreting the generating function for t core partitions in terms of modular forms and then controlling the size of the coefficients using deligne s theorem nee the weil conjectures we also consider congruences for the number of p blocks of sn proving a conjecture of garvan that establishes certain multiplicative congruences when 5 p 23 by using a result of serre concerning the divisibility of coefficients of modular forms we show that for any given prime p and positive integer m the number of p blocks with defect 0 in sn is a multiple of m for almost all n we also establish that any given prime p divides the number of p modularly irreducible representations of sn for almost all n 1996 american mathematical society

on the number of multiplications necessary to compute a length 2 n dft the number of multiplications necessary and sufficient to compute a length 2ndft is determined the method of derivation is shown to apply to the multiplicative complexity results of winograd for a length pndft for p an odd prime number the multiplicative complexity of the one dimensional dft is summarized for many possible lengths

well posedness for the motion of an incompressible liquid with free surface boundary we study the motion of an incompressible perfect liquid body in vacuum this can be thought of as a model for the motion of the ocean or a star the free surface moves with the velocity of the liquid and the pressure vanishes on the free surface this leads to a free boundary problem for euler s equations where the regularity of the boundary enters to highest order we prove local existence in sobolev spaces assuming a physical condition related to the fact that the pressure of a fluid has to be positive

existence and stability of compressible current vortex sheets in three dimensional magnetohydrodynamics compressible vortex sheets are fundamental waves along with shocks and rarefaction waves in entropy solutions to multidimensional hyperbolic systems of conservation laws understanding the behavior of compressible vortex sheets is an important step towards our full understanding of fluid motions and the behavior of entropy solutions for the euler equations in two dimensional gas dynamics the classical linearized stability analysis on compressible vortex sheets predicts stability when the mach number m sqrt 2 and instability when m sqrt 2 for the euler equations in three dimensions every compressible vortex sheet is violently unstable and this instability is the analogue of the kelvin helmholtz instability for incompressible fluids the purpose of this paper is to understand whether compressible vortex sheets in three dimensions which are unstable in the regime of pure gas dynamics become stable under the magnetic effect in three dimensional magnetohydrodynamics mhd one of the main features is that the stability problem is equivalent to a free boundary problem whose free boundary is a characteristic surface which is more delicate than noncharacteristic free boundary problems another feature is that the linearized problem for current vortex sheets in mhd does not meet the uniform kreiss lopatinskii condition these features cause additional analytical difficulties and especially prevent a direct use of the standard picard iteration to the nonlinear problem in this paper we develop a nonlinear approach to deal with these difficulties in three dimensional mhd we first carefully formulate the linearized problem for the current vortex sheets to show rigorously that the magnetic effect makes the problem weakly stable and establish energy estimates especially high order energy estimates in terms of the nonhomogeneous terms and variable coefficients then we exploit these results to develop a suitable iteration scheme of the nash moser hormander type to deal with the loss of the order of derivative in the nonlinear level and establish its convergence which leads to the existence and stability of compressible current vortex sheets locally in time in three dimensional mhd

duhamel hollmann like discrete fourier transform algorithm with the smallest multiplicative complexity over a finite field the new method for the discrete fourier transform computation over a finite field is introduced this method is a nontrivial generalization of the duhamel hollmann algorithm with replacement of the toeplitz convolution calculation by the normalized cyclic convolution calculation both algorithms have the smallest multiplicative complexity

split manageable efficient algorithm for fourier and hadamard transforms in this paper a general efficient manageable split algorithm to compute one dimensional 1 d unitary transforms by using the special partitioning in the frequency domain is introduced the partitions determine fast transformations that split the n point unitary transform into a set of n sub i point transforms i 1 n n sub 1 n sub n n here we introduce a class of splitting transformations the so called paired transforms based on these transforms the decompositions of the fourier transforms of arbitrary orders are given and the corresponding algorithms are considered comparative estimates revealing the efficiency of the proposed algorithms with respect to the known ones are given in particular a proposed method of calculating the 2 sup r point fourier transform requires 2 sup r 1 r 3 2 multiplications and 2 sup r 1 r 9 r sup 2 3r 6 additions in terms of the paired transforms the splitting of the 2 sup r point hadamard transform is described as a result the proposed algorithm for computing this transform uses on the average no more than six operations of additions per sample

a split radix partial input output fast fourier transform algorithm a fast discrete fourier transform dft computing algorithm used in situations where part of the data is zero and only the first transform elements are to be calculated is proposed the method is based on the pruning of a split radix decimation time dit fast fourier transform fft diagram it has the advantage of providing gains as a result of pruning computation and the use of a split radix

on the structure of efficient dft algorithms this paper examines the structure of the prime length discrete fourier transform algorithms that are developed by winograd s approach it is shown that those algorithms have considerable structure and this can be exploited to develop a straightforward design procedure which does not use the chinese remainder theorem and which includes any allowed permutations this structure also allows the design of real data programs and the improvement of the data transfer properties of the prime factor algorithm

recursive cyclotomic factorization a new algorithm for calculating the discrete fourier transform in this paper a new recursive algorithm for calculating the discrete fourier transformation is presented this new so called recursive cyclotomic factorization algorithm rcfa is more efficient than the fast fourier transformation fft algorithm moreover due to its recursive nature the rcfa can also be easily implemented using only a limited number of different computation cells

p theory and modular representation theory in a previous paper 191 we observed a close relationship between the generalized core u g of a finite group g i e roughly speaking the core one has quite often to work with in pure group theory if one leaves the class of p solvable groups and the principal p block it turned out that o g with modifications for p 2 coincides with 0 g the group generated by the support of the centrally primitive idempotent f g corresponding to the principal p block this observation was the central point for a number of new results in modular representation theory which indicate that 0 g may be viewed as the generalized core in modular representation theory from this point of view it is worth paying more attention to this group thus the aim of this note is to polish the theory around o1 g and to see the connection with o g in a new light all groups under consideration are finite all modules finitely generated modules over some group algebra fg where f always denotes a field of characteristic p 0 the author wishes to thank the referee for many helpful suggestions concerning the revision of an earlier version of this paper

the vector split radix algorithm for 2d dht a decimation in frequency vector split radix algorithm is proposed to decompose an n n 2d discrete hartley transform dht into one n 2 n 2 dht and twelve n 4 dhts the proposed algorithm possesses the in place property and needs no matrix transpose its computational structure is very regular and is simpler than those of all existing nonseparable 2d dhts

finite groups whose prime graphs are regular abstract let g be a finite group and let irr g be the set of all irreducible complex characters of g let cd g be the set of all character degrees of g and denote by g the set of primes which divide some character degrees of g the prime graph g associated to g is a graph whose vertex set is g and there is an edge between two distinct primes p and q if and only if the product pq divides some character degree of g in this paper we show that the prime graph g of a finite group g is 3 regular if and only if it is a complete graph with four vertices

the techniques of the generalized fast fourier transform algorithm a general method of deriving dft discrete fourier transform algorithms generalised fast fourier transform algorithms is presented it is shown that a special case of the method is equivalent to nesting of ffts the application of the method to the case where n has mutually prime factors results in a new interpretation of the permutations characteristic of this class of algorithms it is shown that the equalization of ffts leads to results which are different from the widely used intuitive ones the high efficiency of split radix ffts is explained it is shown that the formulae of the method can be easily adapted for deriving algorithms for the cosine sine dft a set of ffts that has smaller arithmetical and or memory complexities than any algorithm known is presented in particular a method of deriving split radix 2 sup s ffts requiring n log sub 2 n 3n 4 real multiplications and 3n log sub 2 n 3n 4 additions for any s 1 is presented

finite group elements where no irreducible character vanishes in this paper we consider elements x of a finite group g with the property that x 0 for all irreducible characters of g if g is solvable and x has odd order we show that x must lie in the fitting subgroup f g

scaled radix 2 8 algorithm for efficient computation of length n 2 m dfts this paper presents a scaled radix 2 8 fast fourier transform fft sr28fft algorithm for computing length n 2m discrete fourier transforms dfts scaled by complex number rotating factors the idea of the sr28fft algorithm is from the modified split radix fft msrfft algorithm and its purpose is to furnish other algorithms with high efficiency but without shortcomings of the msrfft algorithm a novel radix 2 4 fft nr24fft algorithm and a novel radix 2 8 fft nr28fft algorithm are proposed these two algorithms use sr28fft to calculate their sub dfts of odd indexed terms several aspects of the two algorithms such as computational complexity computation accuracy and coefficient evaluations or accesses to the lookup table all are improved the bit reverse method can be used for their order permutation and no extra memory is required to store their extra coefficients by the two novel algorithms which contribute significantly to the performance of the fft algorithms the sr28fft algorithm can also be applied to other algorithms whose decomposition contains sub dfts of powers of two the appendix presents an algorithm named sr28fft 2 for further reducing the number of arithmetic operations and nr24fft and nr28fft algorithms based on sr28fft 2 requires fewer real operations than that required by the msrfft algorithm

well posedness in smooth function spaces for the moving boundary three dimensional compressible euler equations in physical vacuum we prove well posedness for the three dimensional compressible euler equations with moving physical vacuum boundary with an equation of state given by p c for 1 the physical vacuum singularity requires the sound speed c to go to zero as the square root of the distance to the moving boundary and thus creates a degenerate and characteristic hyperbolic free boundary system wherein the density vanishes on the free boundary the uniform kreiss lopatinskii condition is violated and manifest derivative loss ensues nevertheless we are able to establish the existence of unique solutions to this system on a short time interval which are smooth in sobolev spaces all the way to the moving boundary and our estimates have no derivative loss with respect to initial data our proof is founded on an approximation of the euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term chosen to preserve as much of the geometric structure of the euler equations as possible we first construct solutions to this degenerate parabolic regularization using a higher order version of hardy s inequality we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter solutions to the compressible euler equations are found in the limit as the artificial viscosity tends to zero our regular solutions can be viewed as degenerate viscosity solutions our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws

p parts of brauer character degrees abstract let g be a finite group and let p be an odd prime under certain conditions on the p parts of the degrees of its irreducible p brauer characters we prove the solvability of g as a consequence we answer a question proposed by b huppert in 1991 if g has exactly two distinct irreducible p brauer character degrees then is g solvable we also determine the structure of non solvable groups with exactly two irreducible 2 brauer character degrees

a fast algorithm with less operations for length n q times 2 m dfts discrete fourier transform dft is widely used in almost all fields of science and engineering fast fourier transform fft is an efficient tool for computing dft in this paper we present a fast fourier transform fft algorithm for computing length q times 2 m dfts the algorithm transforms all q points sub dfts into three parts in the second part the operations of subtransformation contain only multiplications by real constant factors by transformation length 2 m scaled dfts sdft are obtained an extension of scaled radix 2 8 fft sr28fft is presented for computing these sdfts in which the real constant factors of sdfts are attached to the coefficients of sub dfts to simplify multiplication operations the proposed algorithm achieves reduction of arithmetic complexity over the related algorithms it can achieve a further reduction of arithmetic complexity for computing a length n q times 2 m idft by 2n 4m real multiplications in addition the proposed algorithm is applied to real data fft and is extended to 6 m dfts

on the length of the conjugacy classes of finite groups abstract let g be a finite group the question of how certain arithmetical conditions on the degrees of the irreducible characters of g influence the group structure was studied by several authors our purpose here is to impose analogous conditions on the lengths of the conjugacy classes of g and to describe the group structure under these conditions

implementation of split radix fft algorithms for complex real and real symmetric data a new algorithm is presented for the fast computation of the discrete fourier transform this algorithm belongs to that class of recently proposed 2n fft s which present the same arithmetic complexity the lowest among any previously published one moreover this algorithm has the advantage of being performed in place by repetitive use of a butterfly type structure without any data reordering inside the algorithm furthermore it can easily be applied to real and real symmetric data with reduced arithmetic complexity by removing all redundancy in the algorithm

an answer to a question of isaacs on character degree graphs abstract let n be a normal subgroup of a finite group g we consider the graph g n whose vertices are the prime divisors of the degrees of the irreducible characters of g whose kernel does not contain n and two vertices are joined by an edge if the product of the two primes divides the degree of some of the characters of g whose kernel does not contain n we prove that if g n is disconnected then g n is solvable this proves a strong form of a conjecture of isaacs

fast radix 3 9 discrete hartley transform an efficient algorithm for computing radix 3 9 discrete hartley transforms dhts is presented it is shown that the radix 3 9 fast hartley transform fht algorithm reduces the number of multiplications required by a radix 3 fht algorithm for nearly 50 for the computation of real valued discrete fourier transforms dfts with sequence lengths that are powers of 3 it is shown that the radix 3 9 fht algorithm reduces the number of multiplications by 16 2 over the fastest real valued radix 3 9 fast fourier transform fft algorithm

a new fft algorithm of radix 3 6 and 12 a new algorithm for implementation of radix 3 6 and 12 fft is introduced an fft using this algorithm is computed in an ordinary 1 j complex plane and the number of additions can be significantly reduced the number of multiplication is also reduced high efficiency of the algorithm is derived from the fact that if an input sequence is favorably reordered rotating factors can be treated in pairs so that the rotating factors are conjugate to each other

short time structural stability of compressible vortex sheets with surface tension assume we start with an initial vortex sheet configuration which consists of two inviscid fluids with density bounded below flowing smoothly past each other where a strictly positive fixed coefficient of surface tension produces a surface tension force across the common interface balanced by the pressure jump we model the fluids by the compressible euler equations in three space dimensions with a very general equation of state relating the pressure entropy and density such that the sound speed is positive we prove that for a short time there exists a unique solution of the equations with the same structure

accuracy of the discrete fourier transform and the fast fourier transform fast fourier transform fft based computations can be far more accurate than the slow transforms suggest discrete fourier transforms computed through the fft are far more accurate than slow transforms and convolutions computed via fft are far more accurate than the direct results however these results depend critically on the accuracy of the fft software employed which should generally be considered suspect popular recursions for fast computation of the sine cosine table or twiddle factors are inaccurate due to inherent instability some analyses of these recursions that have appeared heretofore in print suggesting stability are incorrect even in higher dimensions the fft is remarkably stable

a local conjecture on brauer character degrees of finite groups recently a new conjecture on the degrees of the irreducible brauer characters of a finite group was presented by w willems in this paper we propose a local version of this conjecture for blocks b of finite groups giving a lower bound for 1 2 where the sum runs through the set of irreducible brauer characters of b in terms of invariants of b a slight reformulation leads to interesting open questions about traces of cartan matrices of blocks we show that the local conjecture is true for blocks with one simple module blocks of p solvable groups and blocks with cyclic defect groups it also holds for many further examples of blocks of sporadic groups symmetric groups or groups of lie type finally we prove that the conjecture is true for blocks of tame representation type

well posedness for the motion of a compressible liquid with free surface boundary we study the motion of a compressible perfect liquid body in vacuum this can be thought of as a model for the motion of the ocean or a star the free surface moves with the velocity of the liquid and the pressure vanishes on the free surface this leads to a free boundary problem for euler s equations where the regularity of the boundary enters to highest order we prove local existence in sobolev spaces assuming a physical condition related to the fact that the pressure of a fluid has to be positive

on thompson s theorem let p be a prime divisor of the order of a finite group g thompson 1970 j algebra14 129 134 has proved the following remarkable result a finite group g is p nilpotent if the degrees of all its nonlinear irreducible characters are divisible by p in fact in that case g is solvable in this note we prove that a group g having only one nonlinear irreducible character of p degree is a cyclic extension of thompson s group this result is a consequence of the following theorem a nonabelian simple group possesses two nonlinear irreducible characters 1 and 2 of distinct degrees such that p does not divide 1 1 2 1 here p is arbitrary but fixed our proof depends on the classification of finite simple groups some properties of solvable groups possessing exactly two nonlinear irreducible characters of p degree are proved some open questions are posed

on the fixed point accuracy analysis of fft algorithms in this paper we investigate the effect of fixed point arithmetics with limited precision for different fast fourier transform fft algorithms a matrix representation of error propagation model is proposed to analyze the rounding effect an analytic expression of overall quantization loss due to the arithmetic quantization errors is derived to compare the performance with decimation in time dit and decimation in frequency dif configurations from the simulation results the radix 2 dit fft algorithm has better accuracy in term of signal to quantization noise ratio sqnr based on the results a simple criterion of wordlength optimization is proposed to yield comparable accuracy with fewer bit budget

on computing the discrete hartley transform the discrete hartley transform dht is a real valued transform closely related to the dft of a real valued sequence bracewell has recently demonstrated a radix 2 decimation in time fast hartley transform fht algorithm in this paper a complete set of fast algorithms for computing the dht is developed including decimation in frequency radix 4 split radix prime factor and winograd transform algorithms the philosophies of all common fft algorithms are shown to be equally applicable to the computation of the dht and the fht algorithms closely resemble their fft counterparts the operation counts for the fht algorithms are determined and compared to the counts for corresponding real valued fft algorithms the fht algorithms are shown to always require the same number of multiplications the same storage and a few more additions than the real valued fft algorithms even though computation of the fht takes more operations in some situations the inherently real valued nature of the discrete hartley transform may justify this extra cost

extended overlap add and save methods for multirate signal processing the overlap add method ola and overlap save method ols are well known as efficient schemes for high order fir filtering in this correspondence new sampling rate conversion methods are proposed by extending the ola and ols and eliminating the redundancy caused by the conversion first for finite duration sequences a rate conversion with the dft domain approach is discussed then using the result the extended ola and ols are proposed for infinite duration sequences last the computational complexities of our proposed methods are shown

number theoretic transforms over the golden section quadratic field a new number theoretic transform ntt over the real quadratic field q spl radic 5 is suggested and analyzed conventional ntts are used for fast convolution of integer sequences a new approach for computing number theoretic transforms ntts is proposed allowing real signals to be processed as well the method is based on a diophantine approximation of the input real signal before the ntt the choice of the three parameters characterizing any ntt modulus transform length and primitive element is discussed in detail from a practical point of view the suggested ntts offer attractive combinations of these parameters much care has been exercised to reduce the computational complexity the practical usefulness of an irrational number system is demonstrated extensions and open problems are discussed

on the orders of zeros of irreducible characters abstract let g be a finite group and p a prime number we say that an element g in g is a vanishing element of g if there exists an irreducible character of g such that g 0 the main result of this paper shows that if g does not have any vanishing element of p power order then g has a normal sylow p subgroup also we prove that this result is a generalization of some classical theorems in character theory of finite groups

split radix algorithms for length p sup m dft s the split radix algorithm for the discrete fourier transform dft of length 2 sup m is considered first the reason why the split radix algorithm is better than any single radix algorithm on length 2 sup m dfts is given then the split radix approach is generalized to length p sup m dfts it is shown that whenever a radix p sup 2 outperforms a radix p algorithm then a radix p p sup 2 algorithm will outperform both of them as an example a radix 3 9 algorithm is developed for length 3 sup m dfts

groups whose prime graphs have no triangles abstract let g be a finite group and let cd g be the set of all complex irreducible character degrees of g let g be the set of all primes which divide some character degree of g the prime graph g attached to g is a graph whose vertex set is g and there is an edge between two distinct primes u and v if and only if the product uv divides some character degree of g in this paper we show that if g is a finite group whose prime graph g has no triangles then g has at most 5 vertices we also obtain a classification of all finite graphs with 5 vertices and having no triangles which can occur as prime graphs of some finite groups finally we show that the prime graph of a finite group can never be a cycle nor a tree with at least 5 vertices

parametric representation of anatomical structures of the human body by means of trigonometric interpolating sums an approach for the parametric representation of anatomical structures of the human body by means of trigonometric interpolating sums tis is introduced this representation is constructed on the basis of the geometric information provided by medical digital images and an arbitrarily chosen system of curvilinear coordinates the parameterization defined by these coordinates is approximated through tis by using a multidimensional extension of the lancozos s method for accelerating the convergence of trigonometric approximations for smooth nonperiodic functions this allows us to obtain accurate representations for a wide class of anatomical structures including nonclosed ones an upper bound of the approximation error is derived in the sense of the supremum norm the reconstruction of a human face and the surface of a brain cortex are shown as illustrative examples of the parameterization by means of tis

mechanical derivation of fused multiply add algorithms for linear transforms several computer architectures offer fused multiply add fma also called multiply and accumulate mac instructions that are as fast as a single addition or multiplication for the efficient implementation of linear transforms such as the discrete fourier transform or discrete cosine transforms this poses a challenge to algorithm developers as standard transform algorithms have to be manipulated into fma algorithms that make optimal use of fma instructions we present a general method to convert any transform algorithm into an fma algorithm the method works with both algorithms given as directed acyclic graphs dags and algorithms given as structured matrix factorizations we prove bounds on the efficiency of the method in particular we show that it removes all single multiplications except at most as many as the transform has outputs we implemented the dag based version of the method and show that we can generate many of the best known hand derived fma algorithms from the literature as well as a few novel fma algorithms

on linear groups of degrees at most p 1 abstract in this paper we determine the structure of complex linear groups g of degree at most p 1 where p is a t i sylow p subgroup of g

on p parts of character degrees and conjugacy class sizes of finite groups abstract let g be a finite group and irr g the set of irreducible complex characters of g let e p g be the largest integer such that p e p g divides 1 for some irr g we show that g f g p p k e p g for a constant k this settles a conjecture of a moreto 13 conjecture 4 we also study the related problems of the p parts of conjugacy class sizes of finite groups

wiener hopf and spectral factorization of real polynomials by newton s method abstract we present a new method for factoring a real polynomial into the product of two polynomials which have their zeros inside and outside the unit circle respectively the approach is based on solving a nonlinear system by newton s method the jacobi matrix is a polynomial of a companion matrix and thus a bezoutian times the inverse of a triangular toeplitz matrix after getting deeper understanding of the displacement structure of the bezoutian each newton step can be reduced to a few applications of discrete fourier or cosine transforms and the lu decomposition of a conceived linear system with a cauchy like matrix these structural features are employed to design a fast algorithm which shows excellent numerical behavior for example in the case of the extremely ill conditioned test polynomial 2 z n k 0 n 1 z k 2 k 1 n z k of the degree 2 n 20000 we obtain the factorization after 22 newton steps with an error of 3 4 10 8 in the euclidean norm the execution time on a laptop being a couple of minutes the local quadratic convergence of our newton method is proved and explicit convergence balls are given on the basis of estimates for the lipschitz constant which occurs in kantorovich s theorem we also design and test two superfast versions of our method which for polynomials of degree 20000 typically need about half a minute to produce an initial coefficient vector and then perform the newton iteration within 1 s

2 d and 1 d multipaired transforms frequency time type wavelets a concept of multipaired unitary transforms is introduced these kinds of transforms reveal the mathematical structure of fourier transforms and can be considered intermediate unitary transforms when transferring processed data from the original real space of signals to the complex or frequency space of their images considering paired transforms we analyze simultaneously the splitting of the multidimensional fourier transform as well as the presentation of the processed multidimensional signal in the form of the short one dimensional 1 d signals that determine such splitting the main properties of the orthogonal system of paired functions are described and the matrix decompositions of the fourier and hadamard transforms via the paired transforms are given the multiplicative complexity of the two dimensional 2 d 2 sup r spl times 2 sup r point discrete fourier transform by the paired transforms is 4 sup r 2 r 7 3 8 3 12 r 3 which shows the maximum splitting of the 5 d fourier transform into the number of the short 1 d fourier transforms the 2 d paired transforms are not separable and represent themselves as frequency time type wavelets for which two parameters are united frequency and time the decomposition of the signal is performed in a way that is different from the traditional haar system of functions

lie type like groups abstract a finite simple group of lie type in defining characteristic p has exactly two p blocks the principal block and a block of defect zero consisting of the steinberg character whose degree is the p part of the order of the group in this paper we characterize finite groups g which have exactly the principal p block and a p block of defect zero consisting of an irreducible character of degree g p

well posedness of the free surface incompressible euler equations with or without surface tension we provide a new method for treating free boundary problems in perfect fluids and prove local in time well posedness in sobolev spaces for the free surface incompressible 3d euler equations with or without surface tension for arbitrary initial data and without any irrotationality assumption on the fluid this is a free boundary problem for the motion of an incompressible perfect liquid in vacuum wherein the motion of the fluid interacts with the motion of the free surface at highest order

quantization errors in the computation of the discrete hartley transform the principal objective of this paper is the study of the arithmetic roundoff error characteristics of several discrete hartley transform dht algorithms we first summarize a variety of efficient dht algorithms including bracewell s original decimation in time radix 2 algorithm statistical models for fixed and floating point arithmetic roundoff errors are then used as the basis for a theoretical study of roundoff noise characteristics of a number of the dht algorithms the results of a detailed experimental study of roundoff noise are compared to the theoretical predictions in fixed point implementation of the decimation in time and frequency radix 2 algorithms it is found that the noise to signal ratio increases approximately 1 1 bits per stage for the floating point implementation the number of bits of rms noise to signal ratio for all the algorithms increase as sqrt log 2 n so that doubling the number of points produced a mild increase in the output noise

nonsolvable groups satisfying the prime power hypothesis let g be a finite group with the degree set cd g cd g of its complex irreducible characters we call that g satisfies the prime power hypothesis if for distinct degrees 1 1 cd g 1 1 cd g the greatest common divisor gcd 1 1 gcd 1 1 is a prime power in this paper we show that cd g 18 cd g 18 if g is a nonsolvable group satisfying the prime power hypothesis

a new set of minimum add small n rotated dft modules abstract a self sorting in place prime factor fft algorithm introduced in an earlier paper was based on a set of modules for performing small discrete fourier transforms dfts with an optional rotation of the results the modules were designed to reduce the total number of additions in the fft algorithm in specializing this algorithm to the case of real or conjugate symmetric input data it was found necessary to redesign these modules to impose a particular structure in some cases the new modules proved to require fewer operations than the old we describe the design procedure and compare the new operation counts with those for winograd s dft modules the algorithms for the new modules are given in detail these will be useful in coding fft algorithms of either the prime factor or conventional variety

fixed point analysis and parameter selections of msr cordic with applications to fft designs mixed scaling rotation msr coordinate rotation digital computer cordic is an attractive approach to synthesizing complex rotators this paper presents the fixed point error analysis and parameter selections of msr cordic with applications to the fast fourier transform fft first the fixed point mean squared error of the msr cordic is analyzed by considering both the angle approximation error and signal round off error incurred in the finite precision arithmetic the signal to quantization noise ratio sqnr of the output of the fft synthesized using msr cordic is thereafter estimated based on these analyses two different parameter selection algorithms of msr cordic are proposed for general and dedicated msr cordic structures the proposed algorithms minimize the number of adders and word length when the sqnr of the fft output is constrained design examples show that the fft designed by the proposed method exhibits a lower hardware complexity than existing methods

split vector radix fast fourier transform the split radix approach for computing the discrete fourier transform dft is extended for the vector radix fast fourier transform fft to two and higher dimensions it is obtained by further splitting the n 2 n 2 transforms with twiddle factors in the radix 2 2 fft algorithm the generalization of this split vector radix fft algorithm to higher radices and higher dimensions is also presented by introducing a general approach for constructing the fast hartley transform fht from the corresponding fft new vector and split vector radix fht algorithms with the same desirable properties as their fft counterparts are obtained

a general class of split radix fft algorithms for the computation of the dft of length 2 m in this paper a general class of split radix fast fourier transform fft algorithms for computing the length 2m dft is proposed by introducing a new recursive approach coupled with an efficient method for combining the twiddle factors this enables the development of higher split radix fft algorithms from lower split radix fft algorithms without any increase in the arithmetic complexity specifically an arbitrary radix 2 2s fft algorithm for any value of s 4les sles m is proposed and its arithmetic complexity analyzed it is shown that the number of arithmetic operations multiplications plus additions required by the proposed radix 2 2s fft algorithm is independent of s and is 2m 3 2m 1 8 regardless of whether a complex multiplication is carried out using four multiplications and two additions or three multiplications and three additions this paper thus provides a variety of choices and ways for computing the length 2m dft with the same arithmetic complexity

on the motion of vortex sheets with surface tension in three dimensional euler equations with vorticity the motion of vortex sheets with surface tension has been analyzed in the setting of irrotational flows by ambrose 1 and ambrose and masmoudi 2 in two dimensions and by ambrose and masmoudi 3 in three dimensions with irrotationality the nonlinear euler equations reduce to poisson s equation for the pressure function in the bulk and the motion of the vortex sheet is decoupled from that of the fluid thus allowing boundary integral methods to be employed in a general flow with vorticity the full two phase euler equations must be analyzed in this instance the motion of the two phases of fluid is coupled to the motion of the vortex sheet and entirely new mathematical methods must be developed to obtain a well posedness theory in particular a new class of approximation schemes must be employed that preserve the transport type structure of the vorticity an issue that by definition does not arise either in the irrotational theory or in the analysis of the euler equations on fixed domains in the general case with vorticity present in the fluid the vortex sheet is a surface of discontinuity propagated by the fluid representing the material interface between two incompressible inviscid fluids with densities c and respectively the tangential velocity of the fluid suffers a jump discontinuity along the material interface leading to the well known kelvin helmholtz or rayleigh taylor instabilities when surface tension is neglected the velocity of the vortex sheet is the normal component of the fluid velocity whose continuity across the material interface t is enforced in addition to incompressibility the continuity of the normal rather than tangential component of velocity across t is a fundamental difference between vortex sheet evolution and multi d shock wave evolution