branching rules for the supergroup su n m from those of su n m the decomposition of representations of supergroups into representations of subgroups is needed in practical applications in this paper we set up and exploit a fruitful one to one correspondence between the lie group branching su n m su n su m u 1 and the supergroup branchings su n m su n su m u 1 and su n1 n2 m1 m2 su n1 m1 su n2 m2 u 1 a simple and useful prescription is discovered for obtaining the su n m branching rules from those of su n m for any representation a large class of examples sufficient for many physical applications we can foresee are explicitly worked out and tabulated

application of generating function techniques to lie superalgebras the construction of generating functions for multiplicities of irreducible representations of lie superalgebras from generating functions for characters is examined and applied in order to obtain polynomial tensors and branching rules problems arising because of the existence of typical and atypical representations are discussed in detail the techniques are applied to osp 1 2 spl 1 2 osp 3 2 and osp 4 2

even sectors of lie superalgebras as locally convex lie algebras let g be a finite dimensional lie superalgebra over a graded commutative local augmented arens michael ground algebra a lie group is associated with a number of pleasant properties to the even sector g0 viewed as an ordinary complete locally convex lie algebra which is in general infinite dimensional

canonical and differential realizations of osp 3 2 superalgebra the osp 3 2 superalgebra is realized first by means of two supervectorial operators canonically conjugated and then by differential operators acting on functions defined on e 3 2 as for the vectorial representation of osp 1 2 the eigenvalue problem yields to the superspherical harmonics

transformations of some induced osp 3 2 modules in an so 3 sp 2 basis the modules of the orthosymplectic lie superalgebra osp 3 2 induced from finite dimensional irreducible submodules of the stability subalgebra so 3 gl 1 are investigated the corresponding infinite dimensional irreducible or indecomposable modules the kac modules and the related typical and atypical modules are studied in detail every such module is decomposed into a direct sum of either indecomposable or irreducible modules of the even subalgebra so 3 sp 2 for each of these infinite dimensional or finite dimensional irreducible or indecomposable modules relations are written down giving the transformations of the basis under the action of the algebra generators

super toda theories and w algebras from superspace wess zumino witten models using a superspace approach it is proved that an 1 super toda theory can be seen as a constrained wzw model based on a supergroup the gauge transformations which survive the constraints are then useda la drinfeld sokolov to determine explicity the super w algebra underlying this theory the conformal spin content of any such super walgebra is provided in the general case

racah type expressions for the 6j coefficients of the orthosymplectic superalgebra osp 1 2 using generating functions various expressions for the 6j coefficients of the osp 1 2 superalgebra are derived some of these expressions bear a close resemblance to the racah formula for the su 2 6j coefficients as a consequence it is shown that the osp 1 2 6j coefficients exhibit regge symmetries

new single error correcting codes a matrix construction of nonlinear error correcting codes is considered it is shown how this construction and some related theorems can be applied to old codes to get new codes with minimum distance 3 in total 13 new binary single error correcting codes of length at most 511 are obtained

a conjectured character formula for atypical irreducible modules of the lie superalgebra sl m n a formula is used for the character of each finite dimensional irreducible representation of the lie superalgebra sl m n this new formula appears to cover not only typical but also both singly and multiply atypical representations a comparison is made with previous conjectures and the problem of generalising the formula to cover all of the basic classical lie superalgebras is discussed

branching rules for a class of typical and atypical representations of gl m n the irreducible representations of the lie superalgebra gl m n with highest weights of the form 1 2 m are investigated using a recently introduced induced module construction for atypical modules the gl m n gl m n 1 branching rules are obtained and a suitable gel fand tsetlin basis is introduced the class of representations considered includes some multiply atypical irreducible representations of gl m n and all irreducible representations of gl m 1

simple and robust binary self location patterns a simple method to generate a 2 d binary grid pattern which allows for absolute and accurate self location in a finite planar region is proposed the pattern encodes position information in a local way so that reading a small number of its black or white pixels at any place provides sufficient data from which the location can be decoded both efficiently and robustly

the generalized gilbert varshamov bound is implied by turan s theorem code construction the generalization of the gilbert varshamov bound due to gu and fuja 1993 is a direct consequence of turan s theorem on the existence of a clique in a graph with many edges turan s theorem allows a slight improvement of gu and fuja s result this improved generalized gilbert varshamov bound is in fact equivalent to turan s theorem

character formulas for irreducible modules of the lie superalgebras sl m n kac distinguished between typical and atypical finite dimensional irreducible representations of the lie superalgebras sl m n and provided an explicit character formula appropriate to all the typical representations here the range of validity of some character formulas for atypical representations that have been proposed are discussed several of them are of the kac weyl type but then it is proved that all formulas of this type fail to correctly give the character of one particular atypical representation of sl 3 4 having ruled out therefore all such formulas a completely new extension of the kac weyl character formula is proposed the validity of this formula in the case of all covariant tensor irreducible representations is proved and some evidence in support of the conjecture that it covers all irreducible representations of sl m n is presented

on the generalization of molien functions to supergroups the construction of generating functions for the multiplicities of the irreducible representation gamma in the decomposition of sth supersymmetrical power gamma s where gamma and gamma are the representations of some supergroup g is discussed a number of examples are considered the definitions are given for general groups but the derived results are for the simplest specific cases

constructions for perfect maps and pseudorandom arrays a construction of perfect maps i e periodic r v binary arrays in which each n m binary matrix appears exactly once is given a similar construction leads to arrays in which only the zero n m matrix does not appear and to a construction in which only a few n m binary matrices do not appear a generalization to the nonbinary case is given the constructions involve an interesting problem in shift register theory the solution is given for almost all the case of this problem

branching rules for typical and atypical representations of gl n 1 the gl n 1 down arrow gl n gl 1 branching rules are determined for all finite dimensional irreducible typical and atypical representations of gl n mod 1 using a recently introduced induced module construction for atypical modules and confirm those found recently by palev 1988 by another method the lowest weights and characters of the irreducible representations are found and the validity for gl n mod 1 of the character formula conjectured by hughes and king 1987 is confirmed some generalisations and extensions of the method are discussed

the super rotation racah wigner calculus revisited it is shown that since the finite dimensional representations of the super rotation algebra are characterized by the superspin j and the parity of the representation space all features of the racah wigner calculus clebsch gordan coefficients recoupling coefficients as well as the wigner and racah symbols depend on both j and however it is noticed that the dependence on the parities of the wigner and racah symbols can be factorized out into phases so that one can define parity independent super s3 j and s6 j symbols the properties of these symbols are analyzed in particular it is shown that the s6 j symbols possess a symmetry similar to the regge symmetry satisfied by the rotation 6 j symbols analytical and numerical tables of the symbols are given for the lowest values of their arguments

classification of all star and grade star irreps of gl n 1 a method for inducing nondegenerate forms on irreducible gl m n modules that implies some general results on star and grade star representations is investigated these results are applied to obtain a complete classification in terms of highest weights of the irreducible star and grade star representations of gl n 1 it is demonstrated that while gl n 1 admits a large class of star representations the irreducible grade star representations are comparatively rare moreover for n 2 all grade star irreducible representations are also star representations and for n 2 are atypical the superalgebra gl 2 1 proves to be a special case and admits a two parameter family of four dimensional typical grade star irreducible representations that are not star representations in particular typical grade star irreducible representations of gl n 1 exist only for n 1 2

supersymmetries of the superspace e 3 2 the superspace e 3 2 is the simplest superspace that generalizes the three dimensional ordinary space a super euclidean metric is defined on this superspace through its isomorphy with the vectorial representation superspace of the super rotation algebra osp 1 2 the supergroup uosp 3 2 is defined as the largest supersymmetry group acting in e 3 2 and preserving the scalar product the vectorial representation of the superrotation group uosp 1 2 acts also in e 3 2 the relations between these two supersymmetries are studied exhaustively

learning recursive functions a survey studying the learnability of classes of recursive functions has attracted considerable interest for at least four decades starting with gold s 1967 model of learning in the limit many variations modifications and extensions have been proposed these models differ in some of the following the mode of convergence the requirements intermediate hypotheses have to fulfill the set of allowed learning strategies the source of information available to the learner during the learning process the set of admissible hypothesis spaces and the learning goals a considerable amount of work done in this field has been devoted to the characterization of function classes that can be learned in a given model the influence of natural intuitive postulates on the resulting learning power the incorporation of randomness into the learning process the complexity of learning among others on the occasion of rolf wiehagen s 60th birthday the last four decades of research in that area are surveyed with a special focus on rolf wiehagen s work which has made him one of the most influential scientists in the theory of learning recursive functions

on the theory of one parameter subgroups of supergroups the one parameter subgroups of supergroups are defined and their primary properties are delineated parametrization schemes analogous to those used in lie group theory are investigated in particular exponentiation is introduced and related to the one parameter subgroups canonical coordinates of three kinds are investigated the third of which has no analog in conventional lie theory we illustrate our results with the examples of the supersymmetric quantum mechanical superalgebra sqm 2 and the simple superalgebra osp 1 2

chern simons forms on principal superfiber bundles a graded weil homomorphism is defined for principal superfiber bundles and the related transgression or chern simons forms are introduced as an example of the application of these concepts a superextension of the dirac monopole is discussed

classification of star and grade star representations of c n 1 the two types of and grade representations of the lie superalgebra c n 1 are classified a type 1 irreducible representation is characterized by the single condition s 0 and s being the highest weight and the odd simple root respectively while the type 2 representations are duals of type 1 representations for n 1 only the identity and vector representations of c n 1 are shown to be grade the superalgebra c 2 proves to be an exception and admits several classes of nontrivial irreducible grade representations

equivalence of two character formulae for typical and singly atypical irreducible modules of sl m n the bernstein leites van der jeugt character formula known to be valid for all typical and singly atypical irreducible modules of sl m n is shown to be equivalent in all such cases to the formula conjectured by hughes and king 1987 which previously has only been proved valid in the case of sl m 1

on the role of procrastination in machine learning the traditional model of inductive inference is enhanced to allow learning machines to procrastinate about how many trials they will need to complete an inference or about how accurate their solution will be hierarchies of classes of learnable phenomena represented by sets of recursive functions isomorphic to the structure of the constructive ordinals are revealed the existence of such hierarchies indicates a potential advantage of procrastination as a learning technique tradeoffs between the two types of procrastination are shown not to exist in general one of the new classes of sets of inferrible functions introduced in this paper turns out to have the somewhat unusual property of being closed under finite unions a new technique based on a hardest to lear set relative to a class is used

representations of orthosymplectic superalgebras ii young diagrams and weight space techniques for pt i see ibid vol 16 p 473 1983 finite dimensional graded tensor representations of osp m n are enumerated via standard young diagrams and their corresponding highest weight and kac dynkin labels are given a uniform set of conditions on the diagram shape necessary and sufficient for atypicality are presented weight space techniques are used to provide a complete analysis of an atypical representations for the low rank cases osp 2 2 osp 3 2 and osp 4 2

on the composition factors of kac modules for the lie superalgebras sl m n in the classification of finite dimensional modules of lie superalgebras kac distinguished between typical and atypical modules kac introduced an induced module the so called kac module v with highest weight which was shown to be simple if is a typical highest weight if is an atypical highest weight the kac module is indecomposable and the simple module v can be identified with a quotient module of v in the present paper the problem of determining the composition factors of the kac modules for the lie superalgebra sl m n is considered an algorithm is given to determine all these composition factors and conversely an algorithm is given to determine all the kac modules containing a given simple module as a composition factor the two algorithms are presented in the form of conjectures and illustrated by means of detailed examples strong evidence in support of the conjectures is provided the combinatorial way in which the two algorithms are intertwined is both surprising and interesting and is a convincing argument in favor of the solution to the composition factor problem presented here

upper bounds on the rate of ldpc codes as a function of minimum distance new upper bounds on the rate of low density parity check ldpc codes as a function of the minimum distance of the code are derived the bounds apply to regular ldpc codes and sometimes also to right regular ldpc codes their derivation is based on combinatorial arguments and linear programming the new bounds improve upon the previous bounds due to burshtein et al it is proved that at least for high rates regular ldpc codes with full rank parity check matrices have worse relative minimum distance than the one guaranteed by the gilbert varshamov bound

the group with grassmann structure uosp 1 2 the finite dimensional representations of the lie superalgebraosp 1 2 and the group with grassmann structureosp 1 2 have been studied the explicit expression of the projection operator of the superalgebraosp 1 2 has been found the operator permits an arbitrary finite dimensional representation to be expanded in the components multiple to the irreducible ones the clebsch gordan coefficients for the tensor product of two arbitrary irreducible representations have been obtained the matrix elements of the irreducible representations of the groupuosp 1 2 the analoque of the compact form of the grouposp 1 2 are studied the explicit form of these matrix elements the differential equations satisfied by them and the integral of their product have been found

racah wigner calculus for the super rotation algebra i the symmetry properties and the pseudo orthogonality relations of the super rotation clebsh gordan coefficients for the tensor product of two irreducible representations of the super rotation algebra are derived the symmetric super rotation 3 j symbol and the symmetric and invariant super rotation 6 j symbol are defined their basic properties are described and their relations to the usual 3 j and 6 j symbols are given

structure of atypical representations of the lie superalgebras sl m n in contrast to lie algebras lie superalgebras contain typical and atypical irreducible representations character formulae for typical representations of classical lie superalgebras have been known for a long time but the structure of atypical representations remained unsolved the author constructs the character formula for atypical representations of the special linear lie superalgebras sl m n

comparison of identification criteria for machine inductive inference abstract a natural pl 1 hierarchy of successively more general criteria of success for inductive inference machines is described based on the size of sets of anomalies in programs synthesized by such machines these criteria are compared to others in the literature some of our results are interpreted as tradeoff results or as showing the inherent relative computational complexity of certain processes and others are interpreted from a positivistic mechanistic philosophical stance as theorems in philosophy of science the techniques of recursive function theory are employed including ordinary and infinitary recursion theorems

path integrals and supercoherent states for the real supergroup osp 1 2 r with both its compact and noncompact versions supercoherent states are introduced with a method close to the one by perelomov for the even subgroups su 2 or su 1 1 these states labeled by a complex c number and grassmann variable minimize the uncertainty of the quadratic casimir operator of the group a path integral formalism is developed for the transition amplitude between supercoherent states for a hamiltonian linear in the generators of the superalgebra which leads to a super riccati equation finally in the classical limit the canonical equations of motion are derived which involve a generalized super poisson bracket

on the intrinsic complexity of learning a new view of learning is presented the basis of this view is a natural notion of reduction we prove completeness and relative difficulty results an infinite hierarchy of intrinsically more and more difficult to learn concepts is presented our results indicate that the complexity notion captured by our new notion of reduction differs dramatically from the traditional studies of the complexity of the algorithms performing learning tasks

quickest sequence phase detection a phase detection sequence is a length n cyclic sequence such that the location of any length k contiguous subsequence can be determined from a noisy observation of that subsequence in this paper we derive bounds on the minimal possible k in the limit of n to infty and describe some sequence constructions we further consider multiple phase detection sequences where the location of any length k contiguous subsequence of each sequence can be determined simultaneously from a noisy mixture of those subsequences we study the optimal trade offs between the lengths of the sequences and describe some sequence constructions we compare these phase detection problems to their natural channel coding counterparts and show a strict separation between the fundamental limits in the multiple sequence case both adversarial and probabilistic noise models are addressed

properties of clebsch gordan coefficients and 3 j symbols for the quantum superalgebra uq osp 1 mod 2 this paper is a continuation of the study begun in an earlier paper of the structure of grade star representations of the quantum superalgebra uq osp 1 mod 2 the general case of the tensor product of two grade star representations acting in representation spaces with arbitrary not necessarily positive definite hermitian forms is considered an explicit analytical formula for clebsch gordan coefficients for this general case is derived using the projection operator method pseudo orthogonality relations are given and symmetry properties including regge symmetry are discussed the quantum analogues of super 3 j symbols are defined and their symmetry properties are analysed

atypical representations for type i lie superalgebras a new modified induced module construction is presented for all finite dimensional irreducible typical and atypical modules of a type i basic classical lie superalgebra the method is illustrated with some low dimensional representations of gl m mod n and all representations of gl 2 mod 1

clebsch gordan coefficients for the quantum superalgebra uq osp 1 2 the structure of the finite dimensional representations of the quantum superalgebra u q osp 1 2 is considered and the projection operator for this quantum superalgebra is derived application of the projection operator permits one to obtain an explicit analytical formula for the q analog of the clebsch gordan coefficients pseudo orthogonality relations and some other properties of clebsch gordan coefficients are given

on the complexity of inductive inference the notion of the complexity of performing an inductive inference is defined some examples of the tradeoffs between the complexity of performing an inference and the accuracy of the inferred result are presented an axiomatization of the notion of the complexity of inductive inference is developed and several results are presented which both resemble and contrast with results obtainable for the axiomatic computational complexity of recursive functions

path integral for coherent states of the dynamical u2 group and u2 1 supergroup a path integral formulation in the representation of coherent states for the unitary u2 group and u2 1 supergroup is introduced u2 and u2 1 path integrals are shown to be defined on the coset spaces u2 u1 x u1 and u2 1 u1 1 x u1 respectively these cosets appear as curved classical phase spaces partition functions are expressed as path integrals over these spaces in the case when u2 and u2 1 are the dynamical groups the corresponding path integrals are evaluated with the help of linear fractional transformations that appear as the group supergroup action in the coset space superspace possible applications for quantum models are discussed

the capacity region of the totally asynchronous multiple access channel it is shown that the capacity region of the asynchronous multiple access channel differs from that of the synchronous channel only by the lack of a convex hull operation

super angular momentum and super spherical harmonics the usual quantum mechanics methods of defining angular momentum and spherical harmonics in the ordinary space e 3 are generalized to the real commuting super space e 3 2 the super angular momentum generators of the super rotation is realized as differential operators acting on functions defined on e 3 2 by solving the eigenvalue problem in spherical coordinates a set of pseudo orthonormalized functions is constructed the super spherical harmonics ylmj whose main properties are given

diagram and superfield techniques in the classical superalgebras introduces the concept of graded permutation group in the analysis of tensor operators in the classical superalgebras for u m n and su m n irreducible tensor representations correspond to classes of young tableaux with definite graded symmetry type diagram techniques are given for kronecker products dimensions and branching rules the tensor techniques are complemented by the introduction of a superfluid formalism in which u m n and su m n act on polynomial functions over the appropriate superspace such superfluids may admit constraints a general superfield interpolates between the classes of young tableaux which correspond to particular types of constraint the tensor and superfield techniques are illustrated with case studies of su 2 1 and su m 1

rate splitting multiple access for discrete memoryless channels it is shown that the encoding decoding problem for any asynchronous m user discrete memoryless multiple access channel can be reduced to corresponding problems for at most 2m 1 single user discrete memoryless channels this result which extends a similar result for gaussian channels reduces the seemingly hard task of finding good multiple access codes to the much better understood task of finding good codes for single user channels as a by product some interesting properties of the capacity region of m user asynchronous discrete memoryless channels are derived

real forms of uq osp 1 2 and quantum d 2 supersymmetry algebras the authors present three real forms of quantum superalgebra uq osp 1 2 by defining suitable contraction limits they describe the q deformations of d 2 superpoincare and d 2 supereuclidean algebras as hopf bialgebras

perfect maps given positive integers r s u and spl upsi an r s u spl upsi perfect map pm is defined to be a periodic r spl times s binary array in which every u spl times spl upsi binary array appears exactly once as a periodic subarray perfect maps are the natural extension of the de bruijn sequences to two dimensions in the paper the existence question for perfect maps is settled by giving constructions for perfect maps for all parameter sets subject to certain simple necessary conditions extensive use is made of previously known constructions by finding new conditions which guarantee their repeated application these conditions are expressed as bounds on the linear complexities of the periodic sequences formed from the rows and columns of perfect maps

on the intrinsic complexity of learning recursive functions the intrinsic complexity of learning compares the difficulty of learning classes of objects by using some reducibility notion for several types of learning recursive functions both natural complete classes are exhibited and necessary and sufficient conditions for completeness are derived informally a class is complete iff both its topological structure is highly complex while its algorithmic structure is easy some self describing classes turn out to be complete furthermore the structure of the intrinsic complexity is shown to be much richer than the structure of the mind change complexity though in general intrinsic complexity and mind change complexity can behave orthogonally

grassmann integral representation for spanning hyperforests given a hypergraph g we introduce a grassmann algebra over the vertex set and show that a class of grassmann integrals permits an expansion in terms of spanning hyperforests special cases provide the generating functions for rooted and unrooted spanning hyper forests and spanning hyper trees all these results are generalizations of kirchhoff s matrix tree theorem furthermore we show that the class of integrals describing unrooted spanning hyper forests is induced by a theory with an underlying osp 1 2 supersymmetry

channel polarization a method for constructing capacity achieving codes for symmetric binary input memoryless channels a method is proposed called channel polarization to construct code sequences that achieve the symmetric capacity i w of any given binary input discrete memoryless channel b dmc w the symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability channel polarization refers to the fact that it is possible to synthesize out of n independent copies of a given b dmc w a second set of n binary input channels wn i 1 les i les n such that as n becomes large the fraction of indices i for which i wn i is near 1 approaches i w and the fraction for which i wn i is near 0 approaches 1 i w the polarized channels wn i are well conditioned for channel coding one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining codes constructed on the basis of this idea are called polar codes the paper proves that given any b dmc w with i w 0 and any target rate r i w there exists a sequence of polar codes cfrn nges1 such that cfrn has block length n 2n rate ges r and probability of block error under successive cancellation decoding bounded as pe n r les o n 1 4 independently of the code rate this performance is achievable by encoders and decoders with complexity o n logn for each

hook young diagrams with applications to combinatorics and to representations of lie superalgebras definition et etude des tableaux k l semi standards applications a la combinatoire et aux representations des superalgebres de lie des diagrammes de young a crochets