on lie gradings ii abstract to obtain fine gradings of gl n c it is necessary to study maximal abelian subgroups mad subgroups of diagonizable automorphisms in a ut gl n c we described two finite sets h n and k n such that each mad subgroup consisting only from inner automorphisms is conjugated to some element of h n and each mad subgroup containing at least one outer automorphism is conjugated to some element of k n using these results concerning k n we then easily find all mad subgroups in a ut o n c for n 8 and in a ut sp 2n c

deformations and contractions of lie algebras in this contributed presentation we discuss and compare the mutually opposite procedures of deformations and contractions of lie algebras we suggest that with approp riate combinations of both procedures one may construct new lie algebras we first discuss low dimensional lie algebras and illustrate thereby that whereas for every contraction there exists a reverse deformation the converse is not true in general also we note that some lie algebras belonging to parameterized families are singled out by the irreversibility of defor mations and contractions after reminding that global deformations of the witt virasoro and affine kac moody algebras allow one to retrieve lie algebras of krichever novikov type we contract the latter to find new infinite dimensional lie algebras

universality in fully developed turbulence we extend the numerical simulations of she et al phys rev lett 70 3251 1993 of highly turbulent flow with 15 taylor reynolds numbers re lambda 200 up to re lambda approx equals 45 000 employing a reduced wave vector set method introduced earlier to approximately solve the navier stokes equation first also for these extremely high reynolds numbers re lambda the energy spectra as well as the higher moments when scaled by the spectral intensity at the wave number kp of peak dissipation can be described by one universal function of k kp for all re lambda second the k space inertial subrange scaling exponents zeta m of this universal function are in agreement with the 1941 kolmogorov theory the better the larger re lambda is as is the re lambda dependence of kp only around kp viscous damping leads to a slight energy pileup in the spectra as in the experimental data bottleneck phenomenon

permutation presentations of modules over finite groups abstract we introduce a notion of permutation presentations of modules over finite groups and completely determine finite groups over which every module has a permutation presentation to get this result we prove that every coflasque module over a cyclic p group is permutation projective

cartan subalgebras in c algebras existence and uniqueness we initiate the study of cartan subalgebras in c algebras with a particular focus on existence and uniqueness questions for homogeneous c algebras these questions can be analyzed systematically using the theory of fiber bundles for group c algebras while we are able to find cartan subalgebras in c algebras of many connected lie groups there are classes of discrete groups for instance non abelian free groups whose reduced group c algebras do not have any cartan subalgebras moreover we show that uniqueness of cartan subalgebras usually fails for classifiable c algebras however distinguished cartan subalgebras exist in some cases for instance in nuclear uniform roe algebras

graded contractions of the affine lie algebra a1 1 its representations and tensor products and an application to the branching rule a1 1 contains implies a1 1 we describe all contractions of the affine kac moody algebra a1 1 which preserve various gradings by the cyclic groups z2 and z3 as well as a grading by z2 z3 together with the contractions of simultaneously graded representations of a1 1 and their tensor products an application is given to the branching rules for a1 1 contains implies a1 1 for a number of distinct embeddings each associated with a particular grading

contractions of low dimensional lie algebras theoretical background of continuous contractions of finite dimensional lie algebras is rigorously formulated and developed in particular known necessary criteria of contractions are collected and new criteria are proposed a number of requisite invariant and semi invariant quantities are calculated for wide classes of lie algebras including all low dimensional lie algebras an algorithm that allows one to handle one parametric contractions is presented and applied to low dimensional lie algebras as a result all one parametric continuous contractions for both the complex and real lie algebras of dimensions not greater than 4 are constructed with intensive usage of necessary criteria of contractions and with studying correspondence between real and complex cases levels and colevels of low dimensional lie algebras are discussed in detail properties of multiparametric and repeated contractions are also investigated

graded contractions of the gell mann graded sl 3 c the gell mann grading one of the four gradings of sl 3 c that cannot be further refined is considered as the initial grading for the graded contraction procedure using the symmetries of the gell mann grading the system of contraction equations is reduced and solved each non trivial solution of this system determines a lie algebra which is not isomorphic to the original algebra sl 3 c the resulting 53 contracted algebras are divided into two classes the first is represented by the algebras which are also continuous inonu wigner contractions the second is formed by the discrete graded contractions

contractions from grading we note that large classes of contractions of algebras that arise in physics can be understood purely algebraically via identifying appropriate zm gradings and their generalizations on the parent algebra this includes various types of flat space carroll limits of finite and infinite dimensional a ds algebras as well as galilean and galilean conformal algebras our observations can be regarded as providing a natural context for the grassmann approach of krishnan et al j high energy phys 2014 3 36 we also introduce a related notion which we call partial grading that arises naturally in this context

the groupoid of bifractional transformations bifractional transformations which lead to quantities that interpolate between other known quantities are considered they do not form a group and groupoids are used to describe their mathematical structure bifractional coherent states and bifractional wigner functions are also defined the properties of the bifractional coherent states are studied the bifractional wigner functions are used in generalizations of the moyal star formalism a generalized berezin formalism in this context is also studied

mixed modules over zg g cyclic of prime order and over related dedekind pullbacks the structure of all finitely generated zg modules is determined with emphasis on those modules whose additive group is partly torsion and partly torsion free the method is to first find all finitely generated r modules where r is a pullback of two dedekind domains over a field then to specialize to r zg and to other familiar rings

uct kirchberg algebras have nuclear dimension one we prove that every kirchberg algebra in the uct class has nuclear dimension 1 we first show that kirchberg 2 graph algebras with trivial k0 and finite k1 have nuclear dimension 1 by adapting a technique developed by winter and zacharias for cuntz algebras we then prove that every kirchberg algebra in the uct class is a direct limit of 2 graph algebras to obtain our main theorem

the fractional fourier transform and applications this paper describes the fractional fourier transform which admits computation by an algorithm that has complexity proportional to the fast fourier transform algorithm whereas the discrete fourier transform dft is based on integral roots of unity e 2 pi i n the fractional fourier transform is based on fractional roots of unity e 2 pi i alpha where alpha is arbitrary the fractional fourier transform and the corresponding fast algorithm are useful for such applications as computing dfts of sequences with prime lengths computing dfts of sparse sequences analyzing sequences with noninteger periodicities performing high resolution trigonometric interpolation detecting lines in noisy images and detecting signals with linearly drifting frequencies in many cases the resulting algorithms are faster by arbitrarily large factors than conventional techniques

graded contractions of affine lie algebras the method of graded contractions based on the preservation of automorphisms of finite order is applied to the affine lie algebras and their representations to yield a new class of infinite dimensional lie algebras and representations after the introduction of the horizontal and vertical gradings and the algorithm to find the horizontal toroidal gradings some general properties of the graded contractions are discussed and compared with the inonu wigner contractions the example of is discussed in detail

a class of c algebras generalizing both graph algebras and homeomorphism c algebras iv pure infiniteness this is the final one in the series of papers where we introduce and study the c algebras associated with topological graphs in this paper we get a sufficient condition on topological graphs so that the associated c algebras are simple and purely infinite using this result we give one method to construct all kirchberg algebras as c algebras associated with topological graphs

graded contractions of bilinear invariant forms of lie algebra we introduce a new construction for bilinear invariant forms on lie algebras based on the method of graded contractions the general method is described and the z2 z3 and z2 x z2 contractions are found the results can be applied to all lie algebras and superalgebras finite or infinite dimensional which admit the chosen gradings we consider some examples contractions of the killing form toroidal contractions of sl 3 c and we briefly discuss the limit to new wess zumino witten actions

cartan subalgebras and the uct problem ii we study the connection between the uct problem and cartan subalgebras in c algebras the uct problem asks whether every separable nuclear c algebra satisfies the uct i e a noncommutative analogue of the classical universal coefficient theorem from algebraic topology this uct problem is one of the remaining major open questions in the structure and classification theory of simple nuclear c algebras since the class of separable nuclear c algebras is closed under crossed products by finite groups it is a natural and important task to understand the behaviour of the uct under such crossed products we make a contribution towards a better understanding by showing that for certain approximately inner actions of finite cyclic groups on uct kirchberg algebras the crossed products satisfy the uct if and only if we can find cartan subalgebras which are invariant under the actions of our finite cyclic groups we also show that the class of actions we are able to treat is big enough to characterize the uct problem in the sense that every such action even on a particular kirchberg algebra namely the cuntz algebra mathcal o 2 leads to a crossed product satisfying the uct if and only if every separable nuclear c algebra satisfies the uct our results rely on a new construction of cartan subalgebras in certain inductive limit c algebras this new tool turns out to be of independent interest for instance among other things the second author has used it to construct cartan subalgebras in all classifiable unital stably finite c algebras

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

inonu wigner contraction of kac moody algebras inonu wigner contractions of affine kac moody algebras are discussed here the sugawara construction for the contracted affine algebra exists only for a fixed value of the level k which is determined in terms of the dimension of the uncontracted part of the starting lie algebra and the quadratic casimir in the adjoint representation further contractions of the g h coset spaces are discussed and an affine translation algebra is obtained which yields a virasoro algebra via a goddard kent olive gko construction with a central charge given by dim g h

discrete and continuous graded contractions of representations of lie algebras simultaneous grading of lie algebras and their representation spaces is used to develop a new theory of grading preserving contractions of representations of all lie algebras admitting the chosen grading the theory is completely different from the traditional ways of contracting representations the graded contractions fall naturally into two classes discrete and continuous ones related respectively to 2 cocycles and coboundaries of the grading group

discrete and continuous graded contractions of lie algebras and superalgebras grading preserving contractions of lie algebras and superalgebras of any type over the complex number field are defined and studied such contractions fall naturally into two classes the wigner inonu like continuous contractions and new discrete contractions a general method is described for any abelian grading semigroup and any lie algebra or superalgebra admitting such a grading all contractions preserving z2 z3 and z2 z2 gradings are found examples of these gradings and contractions for the simple lie algebra a2 affine kac moody algebra a 1 and the simple superalgebra osp 2 1 are shown

semiprojectivity for certain purely infinite c algebras it is proved that classifiable simple separable nuclear purely infinite c algebras having finitely generated k theory and torsion free k 1 are semiprojective this is accomplished by exhibiting these algebras as c algebras of infinite directed graphs