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

multiple description coding using pairwise correlating transforms the objective of multiple description coding mdc is to encode a source into multiple bitstreams supporting multiple quality levels of decoding in this paper we only consider the two description case where the requirement is that a high quality reconstruction should be decodable from the two bitstreams together while lower but still acceptable quality reconstructions should be decodable from either of the two individual bitstreams this paper describes techniques for meeting mdc objectives in the framework of standard transform based image coding through the design of pairwise correlating transforms the correlation introduced by the transform helps to reduce the distortion when only a single description is received but it also increases the bit rate beyond that prescribed by the rate distortion function of the source we analyze the relation between the redundancy i e the extra bit rate and the single description distortion using this transform based framework we also describe an image coder that incorporates the pairwise transform and show its redundancy rate distortion performance for real images

simple classical groups of lie type are determined by their character degrees let g be a finite group denote by irr g the set of all irreducible complex characters of g let cd g be the set of all irreducible complex character degrees of g forgetting multiplicities that is cd g 1 irr g and let x1 g be the set of all irreducible complex character degrees of g counting multiplicities let h be a finite nonabelian simple classical group in this paper we will show that if g is a finite group and x1 g x1 h then g is isomorphic to h in particular this implies that the nonabelian simple classical groups of lie type are uniquely determined by the structure of their complex group algebras

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

on heights of characters of finite groups abstract in this paper we first state a conjecture on the lower bound of the maximal height of characters in a p block of a finite group then we show that our conjecture holds for all blocks of covering groups of a sporadic simple group for all blocks of a quasi simple group g with g z g isomorphic to a 6 a 7 or a simple group of lie type with an exceptional covering group for all blocks of a symmetric group as well as for all blocks of finite general linear or unitary groups for the proof of the case with symmetric groups it relates to an open question of olsson on the existence of t core partitions where t 3 is an integer as a byproduct our investigation on heights of characters of the symmetric groups and of the general linear or unitary groups also gives evidence for the isaacs moreto navarro tiep conjecture claiming that the number of distinct irreducible character degrees of a sylow p subgroup p of an arbitrary finite group g is at most one more than the number of irreducible character degrees of g that are multiples of p

non vanishing elements of finite groups let g be a finite group and let irr g denote the set of irreducible complex characters of g an element x of g is non vanishing if for every in irr g we have x 0 we prove that if x is a non vanishing element of g and the order of x is coprime to 6 then x lies in the fitting subgroup of g

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

shannon kotel nikov mappings in joint source channel coding this paper deals with lossy joint source channel coding for transmitting memoryless sources over awgn channels the scheme is based on the geometrical interpretation of communication by kotel nikov and shannon where amplitudecontinuous time discrete source samples are mapped directly onto the channel using curves or planes the source and channel spaces can have different dimensions and thereby achieving either compression or error control depending on whether the source bandwidth is smaller or larger than the channel bandwidth we present a general theory for 1 n and m 1 dimension changing mappings and provide two examples for a gaussian source and channel where we optimize both a 2 1 bandwidth reducing and a 1 2 bandwidth expanding mapping both examples show high spectral efficiency and provide both graceful degradation and improvement for imperfect channel state information at the transmitter

formulas for partition k tuples with t cores abstract let a t k n denote the number of partition k tuples of n where each partition is t core in this paper we establish formulas of a t k n for some values of t and k by employing the method of modular forms which extends wang s result for t 3 and k 2 3

sending a laplacian source using hybrid digital analog codes in this paper we study transmission of a memoryless laplacian source over three types of channels additive white laplacian noise awln additive white gaussian noise awgn and slow flat fading rayleigh channels under both bandwidth compression and bandwidth expansion for this purpose we analyze two well known hybrid digital analog hda joint source channel coding schemes for bandwidth compression and one for bandwidth expansion then we obtain achievable absolute error distortion regions of the hda schemes for the matched signal to noise ratio snr case as well as the mismatched snr scenario using numerical examples it is shown that these schemes can achieve a distortion very close to the provided lower bound for the awln channel and to the optimum performance theoretically attainable bound for awgn and rayleigh fading channels on mean absolute error distortion under matched snr conditions in addition a non linear analog coding scheme is analyzed and its performance is compared to the hda schemes for bandwidth compression under both matched and mismatched snr scenarios the results show that the hda schemes outperform the non linear analog coding over the whole csnr region

the heights of irreducible brauer characters in 2 blocks of the symmetric groups abstract we prove that there exists a unique irreducible brauer character of height zero in any 2 blocks of the symmetric group this generalizes the theorem of p fong and g d james that the dimension of every non trivial 2 modular simple module of the symmetric group is even

induction and restriction of characters and hall subgroups abstract let h be a hall subgroup of a separable group g given a character irr g we determine when it happens that g for every irreducible constituent of h and dually given a character irr h we determine when it happens that h for every irreducible constituent of g we also study some related questions

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

self conjugate t core partitions sums of squares and p blocks of an we prove that if t is an integer with t 8 or t 10 then every integer n 2 has a self conjugate t core partition this result has consequences in the representation theory of alternating groups and has a version as a theorem about the representation of integers by sums of squares we also give an infinite sequence of integers that have no self conjugate 9 core partition

some congruences for partitions that are p cores a number of linear congruences modulo r are proved for the number of partitions that are p cores where p is prime 5 p 23 and r is any prime divisor of 1 2 p 1 analogous results are derived for the number of irreducible p modular representations of the symmetric group s n the congruences are proved using the theory of modular forms

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

highest weight theory for truncated current lie algebras abstract in this paper a highest weight theory for the truncated current lie algebra g g g k k t t n 1 k t is developed when the underlying lie algebra g possesses a triangular decomposition the principal result is the reducibility criterion for the verma modules of g for a wide class of lie algebras g including the symmetrizable kac moody lie algebras the heisenberg algebra and the virasoro algebra this is achieved through a study of the shapovalov form

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

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

design of entropy constrained multiple description scalar quantizers the problem of entropy constrained multiple description scalar quantizer design is posed as an optimization problem necessary conditions for optimality are derived and an iterative design algorithm is presented performance results are presented for a gaussian source along with comparisons to the multiple description rate distortion bound and a reference system

joint source channel codes for mimo block fading channels we consider transmission of a continuous amplitude source over an l block rayleigh fading mt x mr multiple input multiple output mimo channel when the channel state information is only available at the receiver since the channel is not ergodic shannon s source channel separation theorem becomes obsolete and the optimal performance requires a joint source channel approach our goal is to minimize the expected end to end distortion particularly in the high signal to noise ratio snr regime the figure of merit is the distortion exponent defined as the exponential decay rate of the expected distortion with increasing snr we provide an upper bound and lower bounds for the distortion exponent with respect to the bandwidth ratio among the channel and source bandwidths for the lower bounds we analyze three different strategies based on layered source coding concatenated with progressive superposition or hybrid digital analog transmission in each case by adjusting the system parameters we optimize the distortion exponent as a function of the bandwidth ratio we prove that the distortion exponent upper bound can be achieved when the channel has only one degree of freedom that is l 1 and min mt mr 1 when we have more degrees of freedom our achievable distortion exponents meet the upper bound for only certain ranges of the bandwidth ratio we demonstrate that our results which were derived for a complex gaussian source can be extended to more general source distributions as well

multiple description wavelet based image coding we consider the problem of coding images for transmission over error prone channels the impairments we target are transient channel shutdowns as would occur in a packet network when a packet is lost or in a wireless system during a deep fade when data is delivered it is assumed to be error free but some of the data may never reach the receiver the proposed algorithms are based on a combination of multiple description scalar quantizers with techniques successfully applied to the construction of some of the most efficient subband coders a given image is encoded into multiple independent packets of roughly equal length when packets are lost the quality of the approximation computed at the receiver depends only on the number of packets received but does not depend on exactly which packets are actually received when compared with previously reported results on the performance of robust image coders based on multiple descriptions on standard test images our coders attain similar psnr values using typically about 50 60 of the bit rate required by these other state of the art coders while at the same time providing significantly more freedom in the mechanism for allocation of redundancy among descriptions

hybrid digital analog hda joint source channel codes for broadcasting and robust communications we consider the problem of transmitting a band limited gaussian source on an additive band limited gaussian noise channel the well known threshold effect dictates that the more powerful a code is the more sensitive it is to the exact knowledge of the channel noise a code is said to be robust if it is asymptotically optimal for a wide range of channel noise thus robust codes have a graceful degradation characteristic and are free of the threshold effect it is demonstrated that robust codes exist whenever the source and channel bandwidths are equal in the unequal bandwidth case a collection of nearly robust joint source channel codes is constructed using a hybrid digital analog hda coding technique for designing nearly robust codes a matched tandem code whose channel encoder s output is partially fully matched to its input is proposed and the existence of an asymptotically optimal matched tandem code is shown the nearly robust codes achieve the shannon limit theoretically optimum distortion and have a less severe threshold effect finally for the case of two different noise conditions the distortion regions of these codes are determined

generalized multiple description coding with correlating transforms multiple description md coding is source coding in which several descriptions of the source are produced such that various reconstruction qualities are obtained from different subsets of the descriptions unlike multiresolution or layered source coding there is no hierarchy of descriptions thus md coding is suitable for packet erasure channels or networks without priority provisions generalizing work by orchard wang vaishampayan and reibman see proc ieee int conf image processing vol i santa barbara ca p 608 11 1997 a transform based approach is developed for producing m descriptions of an n tuple source m spl les n the descriptions are sets of transform coefficients and the transform coefficients of different descriptions are correlated so that missing coefficients can be estimated several transform optimization results are presented for memoryless gaussian sources including a complete solution of the n 2 m 2 case with arbitrary weighting of the descriptions the technique is effective only when independent components of the source have differing variances numerical studies show that this method performs well at low redundancies as compared to uniform md scalar quantization

irreducible induction and nilpotent subgroups in finite groups abstract suppose that g is a finite group and h is a nilpotent subgroup of g if a character of h induces an irreducible character of g then the generalized fitting subgroup of g is nilpotent

characterizing normal sylow p subgroups by character degrees abstract suppose that g is a finite group let p be a prime and let p syl p g we prove that p is normal in g if and only if all the irreducible constituents of the permutation character 1 p g have degree not divisible by p

a variation on theorems of jordan and gluck abstract gluck proved that any finite group g has an abelian subgroup a such that g a is bounded by a polynomial function of the largest degree of the complex irreducible characters of g this improved on a previous bound of isaacs and passman in this paper we present a variation of this result that looks at the number of prime factors all these results in turn may be seen as variations on the classical theorem of jordan on linear groups

on groups of central type a finite group is said to be of central type if it possesses an irreducible complex character which takes the value zero on all noncentral elements equivalently the degree of this character is the square root of the index of the center in 1964 iwahori and matsumoto 15 conjectured that a group of central type must be solvable the paper of liebler and yellen 16 aims to prove this but there is a gap in their proof as explained below nevertheless they do correctly obtain substantial information about a minimal counterexample to the conjecture the aim of this paper is to fill the gap in the liebler yellen proof using as they do the classification of finite simple groups and to provide further information about which solvable groups can have central type we show for instance that every normal subgroup of the central factor group of a solvable group of central type has the property that its system normalizers have square index it is interesting to contrast this restrictive condition with the result of gagola that every solvable group is embeddable in the central factor group of a group of central type see 6 theorem 1 2 the arguments in 16 show that a minimal counterexample g to the iwahori matsumoto conjecture must have the following structure if k z is any minimal normal subgroup of g z where z z g the center then k is abelian k z is a 2 group s cg k is the maximal solvable normal subgroup of g and any minimal normal subgroup his of g s is the direct product of a number l of copies of gl 3 2 indeed the liebler yellen methods can be used to show that the sylow 2 subgroup of k considered as an h s module is a product o f copies of a gl 3 2 module of dimension 4 this module having a unique minimal submodule which is trivial and an irreducible factor module of dimension 3 the minimal submodules are amalgamated in the product

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

multiple description analog joint source channel coding to exploit the diversity in parallel channels the complexity and delay introduced by efficient digital coding strategies may be a barrier in some real time communications in this sense these last years joint source channel coding schemes based on analog mappings have gained prominence precisely for their simplicity and their implicit low delay in this work analog mappings originally designed for point to point communications are adapted to the case of parallel channels by following the multiple description strategy traditionally used in source coding in principle the coding scheme is designed to transmit over parallel awgn on off channels which are characterized by the possibility of having failures we also show that our scheme performs satisfactorily over slow rayleigh fading parallel channels

analog multiple description joint source channel coding based on lattice scaling joint source channel coding schemes based on analog mappings for point to point channels have recently gained attention for their simplicity and low delay in this paper these schemes are extended either to scenarios with or without side information at the decoders to transmit multiple descriptions of a gaussian source over independent parallel channels they are based on a lattice scaling approach together with bandwidth reduction analog mappings adapted for this multiple description scenario the rationale behind lattice scaling is to improve performance through bandwidth expansion another important contribution of this paper is the proof of the separation theorem for the communication scenario of multiple description with side information at the decoders this proof allows us to compare the performance of the proposed schemes with the theoretical limits when considering awgn channels finally when the channels are parallel slow fading rayleigh the proposed schemes are optimal in terms of distortion exponent

achievable rates for multiple descriptions consider a sequence of independent identically distributed i i d random variables x l x 2 cdots x n and a distortion measure d x i x i on the estimates x i of x i two descriptions i x in 1 2 cdots 2 nr 1 and j x in 1 2 cdots 2 nr 2 are given of the sequence x x 1 x 2 cdots x n from these two descriptions three estimates i x x2 j x and hat x o i x j x are formed with resulting expected distortions e frac 1 n sum n k 1 d x k hat x mk d m m 0 1 2 we find that the distortion constraints d 0 d 1 d 2 are achievable if there exists a probability mass distribution p x p hat x 1 hat x 2 hat x 0 x with ed x hat x m leq d m such that r 1 i x hat x 1 r 2 i x hat x 2 where i cdot denotes shannon mutual information these rates are shown to be optimal for deterministic distortion measures

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

explicit formulas for partition pairs and triples with 3 cores abstract let a 3 n resp b 3 n denote the number of partition pairs resp triples of n where each partition is 3 core by applying ramanujan s 1 1 formula and bailey s 6 6 formula we find the explicit formulas for a 3 n and b 3 n using these formulas we confirm a conjecture of xia and establish many arithmetic identities satisfied by a 3 n and b 3 n

compatibility of the feigin frenkel isomorphism and the harish chandra isomorphism for jet algebras let g be a simple finite dimensional complex lie algebra with a cartan subalgebra h and weyl group w let g denote the lie algebra of n jets on g a theorem of rais and tauvel and geoffriau identifies the centre of the category of g modules with the algebra of functions on the variety of n jets on the affine space h w on the other hand a theorem of feigin and frenkel identifies the centre of the category of critical level smooth modules of the corresponding affine kac moody algebra with the algebra of functions on the ind scheme of opers for the langlands dual group we prove that these two isomorphisms are compatible by defining the higher residue of opers with irregular singularities we also define generalized verma and wakimoto modules and relate them by a nontrivial morphism

multiple description distribution preserving quantization the notion of a multiple description quantizer mdq includes providing multiple distortion levels if a human observer is involved the design of mdq requires a suitable distortion measure to achieve a graceful quality degradation in the case of description losses while the mean squared error is a ubiquitous distortion measure for many classic mdq schemes it is known to be perceptually relevant only at low distortions we propose a new mdq designed according to an unconventional distortion criterion that combines the mean squared error with a constraint on the probability distribution of the reconstructed signal the performance of the new mdq is shown to approach that of the classic mdq asymptotically as rate increases however once applied in the context of transform audio coding the new mdq significantly outperforms a classic mdq in perceptual tests the new scheme is suitable for a wide range of distortions and renders a seamless transition between coding that preserves signal features and coding of a waveform

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

deterministic annealing based optimization for zero delay source channel coding in networks this paper studies the problem of global optimization of zero delay source channel codes that map between the source space and the channel space under a given transmission power constraint and for the mean square error distortion particularly we focus on two well known network settings the wyner ziv setting where only a decoder has access to side information and the distributed setting where independent encoders transmit over independent channels to a central decoder prior work derived the necessary conditions for optimality of the encoder and decoder mappings along with a greedy optimization algorithm that imposes these conditions iteratively in conjunction with the heuristic noisy channel relaxation method to mitigate poor local minima while noisy channel relaxation is arguably effective in simple settings it fails to provide accurate global optimization in more complicated settings considered in this paper we propose a powerful nonconvex optimization method based on the concept of deterministic annealing which is derived from information theoretic principles and was successfully employed in several problems including vector quantization classification and regression we present comparative numerical results that show strict superiority of the proposed method over greedy optimization methods as well as prior approaches in literature

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

multiple description vector quantization with lattice codebooks design and analysis the problem of designing a multiple description vector quantizer with lattice codebook spl lambda is considered a general solution is given to a labeling problem which plays a crucial role in the design of such quantizers numerical performance results are obtained for quantizers based on the lattices a sub 2 and z sup i i 1 2 4 8 that make use of this labeling algorithm the high rate squared error distortions for this family of l dimensional vector quantizers are then analyzed for a memoryless source with probability density function pdf p and differential entropy h p spl infin for any a spl epsiv 0 1 and rate pair r r it is shown that the two channel distortion d sub o and the channel 1 or channel 2 distortion d sub s satisfy lim sub r spl rarr spl infin d sub o 2 sup 2r 1 a 1 4 g spl lambda 2 sup 2h p and lim sub r spl rarr spl infin d sub s 2 sup 2r 1 a g s sub l 2 sup 2h p where g spl lambda is the normalized second moment of a voronoi cell of the lattice spl lambda and g s sub l is the normalized second moment of a sphere in l dimensions

on the distortion snr exponent of hybrid digital analog space time coding we consider the transmission of a real independent and identically distributed i i d analog source over a quasi static m input n output multiple input multiple output mimo block fading channel the relevant performance criterion is end to end average quadratic distortion d versus channel signal to noise ratio snr for given spectral efficiency eta defined as the ratio of the source bandwidth over the channel bandwidth in the limit of high snr we define the distortion snr exponent a eta as the largest a such that d esdot snr a over all possible source channel coding schemes of spectral efficiency eta we find a simple upper bound on a eta an achievable lower bound asep eta achievable by separated tandem source channel coding and a tighter lower bound ahybrid eta achievable by new hybrid digital analog space time coding schemes as a corollary we have that a eta is completely determined for the scalar case m n 1 and for the bandwidth compression case eta ges 2 min m n expiicit and simple construction of hybrid space time codes achieving ahybrid eta are also given

cranks and t cores new statistics on partitions calledcranks are defined which combinatorially prove ramanujan s congruences for the partition function modulo 5 7 11 and 25 explicit bijections are given for the equinumerous crank classes the cranks are closely related to thet core of a partition usingq series some explicit formulas are given for the number of partitions which aret cores some related questions for self conjugate and distinct partitions are discussed

on vanishing class sizes in finite groups abstract let g be a finite group an element g of g is called a vanishing element if there exists an irreducible character of g such that g 0 in this case we say that the conjugacy class of g is a vanishing conjugacy class in this paper we discuss some arithmetical properties concerning the sizes of the vanishing conjugacy classes in a finite group

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

explicit bounds for the number of p core partitions in this article we derive explicit bounds on ct n the number of t core partitions of n in the case when t p is prime we express the generating function f z as the sum

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

the largest character degree and the sylow subgroups of finite groups let p be a sylow p subgroup and b g the largest irreducible character degree of a finite group g it is shown that p op g b g in general and p op g b g in case p is abelian

on vanishing criteria that control finite group structure abstract many results have been established that show how arithmetic conditions on conjugacy class sizes affect group structure a conjugacy class in g is called vanishing if there exists some irreducible character of g which evaluates to zero on the conjugacy class the aim of this paper is to show that for some classical results it is enough to consider the same arithmetic conditions on the vanishing conjugacy classes of the group

a generalization of strongly monomial groups abstract olivieri del rio and simon defined strongly monomial groups and a significant result proved by them is the explicit description of the simple components of the rational group algebra q g of a strongly monomial group g in this paper generalized strongly monomial groups are defined which is a generalization of strongly monomial groups beside strongly monomial groups an extensive list of important families of groups which are generalized strongly monomial is produced a description of the simple components of q g when g is a generalized strongly monomial group is also provided furthermore the work by jespers olteanu del rio and van gelder on the construction of a subgroup of finite index in the group of central units of integral group ring of a strongly monomial group has also been generalized

design of multiple description scalar quantizers the design of scalar quantizers for communication systems that use diversity to overcome channel impairments is considered the design problem is posed as an optimization problem and necessary conditions for optimality are derived a design algorithm a generalization of s p lloyd s 1962 algorithm for quantizer design is developed unlike a single channel scalar quantizer the performance of a multiple description scalar quantizer is dependent on the index assignment the problem of index assignment is addressed good index assignments performance results and sample quantizer designs are presented for a memoryless gaussian source comparisons are made with rate distortion bounds for the multiple description problem

formulas for cubic partition with 3 cores abstract let c 3 n denote the number of cubic partitions of n with 3 cores in this paper we establish the arithmetic properties and formulas for c 3 n by employing bailey s 6 6 formula and theta function identities