structural phase transition in evolving networks a network as a substrate for dynamic processes may have its own dynamics we propose a model for networks which evolve together with diffusing particles through a coupled dynamics and investigate emerging structural property the model consists of an undirected weighted network of fixed mean degree and randomly diffusing particles of fixed density the weight w of an edge increases by the amount of traffics through its connecting nodes or decreases by a constant factor edges are removed with the probability p rew 1 1 w and replaced by new ones having w 0 at random locations we find that the model exhibits a structural phase transition between the homogeneous phase characterized by an exponentially decaying degree distribution and the heterogeneous phase characterized by the presence of hubs the hubs emerge as a consequence of a positive feedback between the particle and the edge dynamics

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

scaling behavior of explosive percolation on the square lattice clusters generated by the product rule growth model of achlioptas d souza and spencer on a two dimensional square lattice are shown to obey qualitatively different scaling behavior than standard random growth percolation the threshold with unrestricted bond placement allowing loops is found precisely using several different criteria based on both moments and wrapping probabilities yielding p c 0 526 text 565 ifmmode pm else textpm fi 0 000 text 005 consistent with the recent result of radicchi and fortunato the correlation length exponent ensuremath nu is found to be close to 1 the qualitative difference from regular percolation is shown dramatically in the behavior of the percolation probability p ensuremath infty size of largest cluster of the susceptibility and of the second moment of finite clusters where discontinuities appear at the threshold the critical cluster size distribution does not follow a consistent power law for the range of system sizes we study l ensuremath le 8192 but may approach a power law with ensuremath tau g2 for larger l

on duality of multiobjective fractional measurable subset selection problems in this paper we prove some duality results for the multiobjective fractional measurable subset selection problems under more general assumptions than convexity

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

optimality conditions for multiobjective programming with generalized i ρ θ convex set functions abstract necessary conditions for pareto optimality in multiobjective programming with subdifferentiable set functions are established in theorem 12 of h c lai and l j lin j math anal appl 132 1988 558 571 in this paper we establish some sufficient conditions under which a feasible solution of such a problem will be pareto optimal provided that a weaker convexity requirement is satisfied for instance i convexity is assumed for both objective and constraint set functions some duality models are also discussed wolfe type and mond weir type duality theorems are proved

optimization theory for n set functions abstract an optimization theory is developed for functions of n sets optimality conditions are established and a lagrangian duality is obtained

saddle points and lagrangian type duality for discrete minmax fractional subset programming problems with generalized convex functions abstract in this paper we present four sets of saddle point type optimality conditions construct two lagrangian type dual problems and prove weak and strong duality theorems for a discrete minmax fractional subset programming problem we establish these optimality and duality results under appropriate b convexity hypotheses

optimality and duality for a multi objective programming problem involving generalized d type i and related n set functions in this paper we introduce several generalized convexity for a real valued set function and establish optimality and duality results for a multi objective programming problem involving generalized d type i and related n set functions

on the optimality conditions of vector valued n set functions abstract in a finite atomless measure space x the optimization problem of vector valued n set functions defined on a convex subfamily s of n is investigated the necessary and sufficient conditions of pareto optimal solution or proper r p solution of optimization problem with differentiable vector valued n set functions are given

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

representations of osp m 2n and young supertableaux young supertableaux are introduced for finite dimensional representations of orthosymplectic superalgebras and used to study their content

degree dependent network growth from preferential attachment to explosive percolation we present a simple model of network growth and solve it by writing the dynamic equations for its macroscopic characteristics such as the degree distribution and degree correlations this allows us to study carefully the percolation transition using a generating functions theory the model considers a network with a fixed number of nodes wherein links are introduced using degree dependent linking probabilities pk to illustrate the techniques and support our findings using monte carlo simulations we introduce the exemplary linking rule pk k with between 1 and this parameter may be used to interpolate between different regimes for negative links are most likely attached to high degree nodes on the other hand in case 0 nodes with low degrees are connected and the model asymptotically approaches a process undergoing explosive percolation

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

agglomerative percolation on the bethe lattice and the triangular cactus agglomerative percolation ap on the bethe lattice and the triangular cactus is studied to establish the exact mean field theory for ap using the self consistent simulation method based on the exact self consistent equations the order parameter p and the average cluster size s are measured from the measured p and s the critical exponents k and k for k 2 and 3 are evaluated here k and k are the critical exponents for p and s when the growth of clusters spontaneously breaks the zk symmetry of the k partite graph the obtained values are 2 1 79 3 2 0 88 1 3 1 35 5 and 3 0 94 2 by comparing these exponents with those for ordinary percolation 1 and 1 we also find 3 2 these results quantitatively verify the conjecture that the ap model belongs to a new universality class if the zk symmetry is broken spontaneously and the new universality class depends on k

mixed supertableaux of the superunitary groups ii su n m the young supertableaux of the superunitary groups are studied we give the conditions on the size of legal supertableaux the type of atypicity for tensor representations which are irreducible making the link between the kac dynkin parameters and those of the supertableau a discussion is given on nonfully reducible representations and generalized atypical supertableaux

percolation on correlated random networks we consider a class of random weighted networks obtained through a redefinition of patterns in an hopfield like model and by performing percolation processes we get information about topology and resilience properties of the networks themselves given the weighted nature of the graphs different kinds of bond percolation can be studied stochastic deleting links randomly and deterministic deleting links based on rank weights each mimicking a different physical process the evolution of the network is accordingly different as evidenced by the behavior of the largest component size and of the distribution of cluster sizes in particular we can derive that weak ties are crucial in order to maintain the graph connected and that when they are the most prone to failure the giant component typically shrinks without abruptly breaking apart these results have been recently evidenced in several kinds of social networks

explosive site percolation with a product rule we study the site percolation under achlioptas process with a product rule in a two dimensional square lattice from the measurement of the cluster size distribution p s we find that p s has a very robust power law regime followed by a stable hump near the transition threshold based on the careful analysis on the p s distribution we show that the transition should be discontinuous the existence of the hysteresis loop in order parameter also verifies that the transition is discontinuous in two dimensions moreover we also show that the transition nature from the product rule is not the same as that from a sum rule in two dimensions

percolation on correlated networks we reconsider the problem of percolation on an equilibrium random network with degree degree correlations between nearest neighboring vertices focusing on critical singularities at a percolation threshold we obtain criteria for degree degree correlations to be irrelevant for critical singularities we present examples of networks in which assortative and disassortative mixing leads to unusual percolation properties and new critical exponents

finite dimensional representations of the lie superalgebra sl 1 3 in a gel fand zetlin basis ii nontypical representations in a series of two papers all finite dimensional irreducible representations of the special linear lie superalgebra sl 1 3 are constructed explicit formulas for the generators in an orthonormal gel fand zetlin basis of the even subalgebra gl 3 are given this paper develops a background for constructing the representations expressions for the transformation properties of the basis under the action of the generators are written down within all typical sl 1 3 modules

a new route to explosive percolation abstract the biased link occupation rule in the achlioptas process ap discourages the large clusters from growing much ahead of others and encourages faster growth of clusters which lag behind in this paper we propose a model where this tendency is sharply reflected in the gamma distribution of the cluster sizes unlike the power law distribution in ap in this model single edges between pairs of clusters of sizes s i and s j are occupied with a probability s i s j the parameter is continuously tunable over the entire real axis numerical studies indicate that for c the transition is first order c 0 for a square lattice and c 1 2 for random graphs in the limits of this model coincides with models well established in the literature

percolation model with an additional source of disorder the ranges of transmission of the mobiles in a mobile ad hoc network are not uniform in reality they are affected by the temperature fluctuation in air obstruction due to the solid objects even the humidity difference in the environment etc how the varying range of transmission of the individual active elements affects the global connectivity in the network may be an important practical question to ask here a model of percolation phenomena with an additional source of disorder is introduced for a theoretical understanding of this problem as in ordinary percolation sites of a square lattice are occupied randomly with probability p each occupied site is then assigned a circular disk of random value r for its radius a bond is defined to be occupied if and only if the radii r 1 and r 2 of the disks centered at the ends satisfy a certain predefined condition in a very general formulation one divides the r 1 r 2 plane into two regions by an arbitrary closed curve one defines a point within one region as representing an occupied bond otherwise it is a vacant bond the study of three different rules under this general formulation indicates that the percolation threshold always varies continuously this threshold has two limiting values one is p c sq the percolation threshold for the ordinary site percolation on the square lattice and the other is unity the approach of the percolation threshold to its limiting values are characterized by two exponents in a special case all lattice sites are occupied by disks of random radii r 0 r 0 and a percolation transition is observed with r 0 as the control variable similar to the site occupation probability

duality for a minimax programming problem containingn set functions abstract parameter necessary and sufficient optimality conditions for the discrete minimax programming problem with differentiable n set functions are established in zalmai optimization 20 1989 377 395 in this paper we establish nonparameter necessary and sufficient optimality conditions the nonparameter necessary optimality conditions suggest the establishment of two parameter free duality models for the discrete minimax programming problem with differentiable n set functions and the weak strong and strict converse duality theorems have been derived

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

cluster aggregation model for discontinuous percolation transitions the evolution of the erd ifmmode mbox h o else h o fi s r enyi er network by adding edges is a basis model for irreversible kinetic aggregation phenomena such er processes can be described by a rate equation for the evolution of the cluster size distribution with the connection kernel k ij ensuremath sim ij where ij is the product of the sizes of two merging clusters here we study that when the giant cluster is discouraged to develop by a sublinear kernel k ij ensuremath sim ij ensuremath omega with 0 ensuremath le ensuremath omega l1 2 the percolation transition pt is discontinuous such discontinuous pt can occur even when the er dynamics evolves from proper initial conditions the obtained evolutionary properties of the simple model sheds light on the origin of the discontinuous pt in other nonequilibrium kinetic systems

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

moreau rockafellar type theorem for convex set functions abstract let x be an atomless finite measure space and s a convex subfamily it is proved that the moreau rockafellar theorem f 1 f n f 1 f n holds for proper convex set functions f 1 f n and s if all set functions f i except possibly one are w lower semicontinuous on s as applications the kuhn tucker type condition for an optimal solution of convex programming problem with set functions and the fritz john type condition for an optimal solution of vector valued minimization problem for set functions are obtained

unstable supercritical discontinuous percolation transitions percolation is a pervasive concept 1 which has applications in a wide variety of natural technological and social systems 2 3 4 5 ranging from conductivity of composite materials 6 7 and polymerization 8 to epidemic spreading 9 10 11 and information diffusion 12 13

finite dimensional representations of the lie superalgebra gl 2 2 in a gl 2 gl 2 basis ii nontypical representations all finite dimensional irreducible representations of the general linear lie superalgebra gl 2 2 are written down in matrix form the basis within each representation space is chosen in such a way that it makes evident the decomposition of gl 2 2 into irreducible representations of its even subalgebra gl 2 gl 2 special attention is devoted to the analysis of all nontypical representations and some indecomposible representations

percolation transition in networks with degree degree correlation we introduce an exponential random graph model for networks with a fixed degree distribution and a tunable degree degree correlation we then investigate the nature of the percolation transition in a correlated network with a poisson degree distribution it is found that negative correlation is irrelevant in that the percolation transition in the disassortative network belongs to the same universality class as in the uncorrelated network positive correlation turns out to be relevant the percolation transition in the assortative network is characterized by the nondiverging mean size of finite clusters and power law scalings of the density of the largest cluster and the cluster size distribution in the nonpercolating phase as well as at the critical point our results suggest that the unusual type of percolation transition in the growing network models reported recently may be inherited from the assortative degree degree correlation

efficiency conditions and duality models for multiobjective fractional subset programming problems with generalized ϝ α ϱ θ v convex functions abstract in this paper we first introduce a new class of generalized convex n set functions called v convex functions and then present numerous sets of parametric and semiparametric sufficient efficiency conditions under generalized v convexity assumptions for a multiobjective fractional subset programming problem moreover we construct three parametric and three semiparametric duality models and prove appropriate duality theorems

irreducible representations of the superalgebras type ii the representation of the orthosymplectic algebras and the other members of their class are built explicitly with simple techniques

continuity of the explosive percolation transition the explosive percolation problem on the complete graph is investigated via extensive numerical simulations we obtain the cluster size distribution at the moment when the cluster size heterogeneity becomes maximum the distribution is found to be well described by the power law form with the decay exponent ensuremath tau 2 06 2 followed by a hump we then use the finite size scaling method to make all the distributions at various system sizes up to n 2 37 collapse perfectly onto a scaling curve characterized solely by the single exponent ensuremath tau we also observe that the instant of that collapse converges to a well defined percolation threshold from below as n ensuremath rightarrow ensuremath infty based on these observations we show that the explosive percolation transition in the model should be continuous contrary to the widely spread belief of its discontinuity

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

on minimax fractional programming of generalized convex set functions abstract we derive necessary and sufficient optimality conditions for the discrete minimax programming problem with subdifferentiable i convex set functions then we apply these optimality criteria to construct two models of parameter free dual problems and a third model of another dual problem we also establish weak strong and strictly converse duality theorems

method for estimating critical exponents in percolation processes with low sampling in phase transition phenomena the estimation of the critical point is crucial for the calculation of the various critical exponents and the determination of the universality class they belong to however this is not an easy task since a large amount of realizations is needed to eliminate the noise in the data in this paper we introduce a novel method for the simultaneous estimation of the critical point p c and the critical exponent applied for the case of explosive bond percolation on two dimensional square lattices and erdos renyi networks the results show that with only a few hundred realizations it is possible to acquire accurate values for these quantities guidelines are given at the end for the applicability of the method to other cases as well

infinite order percolation and giant fluctuations in a protein interaction network we investigate a model protein interaction network whose links represent interactions between individual proteins this network evolves by the functional duplication of proteins supplemented by random link addition to account for mutations when link addition is dominant an infinite order percolation transition arises as a function of the addition rate in the opposite limit of high duplication rate the network exhibits giant structural fluctuations in different realizations for biologically relevant growth rates the node degree distribution has an algebraic tail with a peculiar rate dependence for the associated exponent

necessary and sufficient conditions for minimax fractional programming we establish the necessary and sufficient optimality conditions for a class of nondifferentiable minimax fractional programming problems solving generalized convex functions subsequently we apply the optimality conditions to formulate one parametric dual problem and we prove weak duality strong duality and strict converse duality theorems

irreducible representations of the exceptional lie superalgebras d 2 1 α the shift operator technique is used to give a complete analysis of all finite and infinite dimensional irreducible representations of the exceptional lie superalgebrasd 2 1 for all cases the star or grade star conditions for the algebra are investigated among the finite dimensional representations there are no star and only a few grade star representations but an infinite class of infinite dimensional star representations is found explicit expressions are given for the doublet representation of d 2 1 the one missing label problem d 2 1 su 2 su 2 su 2 is discussed in detail and solved explicitly

young supertableaux of the basic lie superalgebras among the simple lie superalgebras lsa the basic classical ones are well known together with many of their finite dimensional representations this is particularly true for the typical representations which are known to be completely reducible it is well known that reducible but indecomposable representations of the basic lsa are not completely characterised by their highest weight hw or the eigenvalues of their casimir operators in fact there may be several different finite representations characterised by the same hw such representations are called non typical or atypical the aim of this paper is to establish a complete characterisation of the typical and non typical representations by means of properly defined young supertableaux yst therefore the definition we are going to use for our yst differs slightly from the standard definitions the examples given throughout this paper help the illustration of this new interpretation for the yst which makes possible the complete classification of the indecom posable representations characterised by a generalised notion of hw

on unitary lie superalgebras from the spin orbit supersymmetrisation procedure from the basis elements of a clifford algebra cl n the authors generate a grading leading to a unitary lie superalgebra when n is even such a construction is motivated by the understanding of the specific properties of the fermionic variables in the so called spin orbit coupling procedure of supersymmetrisation in n 2 supersymmetric quantum mechanics the n odd case is also considered and some specific examples are discussed

cluster size heterogeneity in the two dimensional ising model we numerically investigate the heterogeneity in cluster sizes in the two dimensional ising model and verify its scaling form recently proposed in the context of percolation problems phys rev e 84 010101 r 2011 the scaling exponents obtained via the finite size scaling analysis are shown to be consistent with theoretical values of the fractal dimension d f and the fisher exponent ensuremath tau for the cluster distribution we also point out that strong finite size effects exist due to the geometric nature of the cluster size heterogeneity

finite size scaling theory for explosive percolation transitions the finite size scaling fss theory for continuous phase transitions has been useful in determining the critical behavior from the size dependent behaviors of thermodynamic quantities when the phase transition is discontinuous however fss approach has not been well established yet here we develop a fss theory for the explosive percolation transition arising in the erd s and renyi model under the achlioptas process a scaling function is derived based on the observed fact that the derivative of the curve of the order parameter at the critical point t c diverges with system size in a power law manner which is different from the conventional one based on the divergence of the correlation length at t c we show that the susceptibility is also described in the same scaling form numerical simulation data for different system sizes are well collapsed on the respective scaling functions

are randomly grown graphs really random we analyze a minimal model of a growing network at each time step a new vertex is added then with probability ensuremath delta two vertices are chosen uniformly at random and joined by an undirected edge this process is repeated for t time steps in the limit of large t the resulting graph displays surprisingly rich characteristics in particular a giant component emerges in an infinite order phase transition at ensuremath delta 1 8 at the transition the average component size jumps discontinuously but remains finite in contrast a static random graph with the same degree distribution exhibits a second order phase transition at ensuremath delta 1 4 and the average component size diverges there these dramatic differences between grown and static random graphs stem from a positive correlation between the degrees of connected vertices in the grown graph older vertices tend to have higher degree and to link with other high degree vertices merely by virtue of their age we conclude that grown graphs however randomly they are constructed are fundamentally different from their static random graph counterparts

explosive percolation a numerical analysis percolation is one of the most studied processes in statistical physics a recent paper by achlioptas et al science 323 1453 2009 showed that the percolation transition which is usually continuous becomes discontinuous explosive if links are added to the system according to special cooperative rules achlioptas processes in this paper we present a detailed numerical analysis of achlioptas processes with product rule on various systems including lattices random networks a la erdos renyi and scale free networks in all cases we recover the explosive transition by achlioptas et al however the explosive percolation transition is kind of hybrid as despite the discontinuity of the order parameter at the threshold one observes traces of analytical behavior such as power law distributions of cluster sizes in particular for scale free networks with degree exponent 3 all relevant percolation variables display power law scaling just as in continuous second order phase transitions

continuous percolation phase transitions of random networks under a generalized achlioptas process using finite size scaling we have investigated the percolation phase transitions of evolving random networks under a generalized achlioptas process gap during this gap the edge with a minimum product of two connecting cluster sizes is taken with a probability p from two randomly chosen edges this model becomes the erdos renyi network at p 0 5 and the random network under the achlioptas process at p 1 using both the fixed point of the size ratio s 2 s 1 and the straight line of ln s 1 where s 1 and s 2 are the reduced sizes of the largest and the second largest cluster we demonstrate that the phase transitions of this model are continuous for 0 5 0 9 beta upsilon and s 2 s 1 at critical point vary with p and the universality class of phase transitions depends on p

explosive site percolation and finite size hysteresis we report the critical point for site percolation for the explosive type for two dimensional square lattices using monte carlo simulations and compare it to the classical well known percolation we use similar algorithms as have been recently reported for bond percolation and networks we calculate the explosive site percolation threshold as p c 0 695 and we find evidence that explosive site percolation surprisingly may belong to a different universality class than bond percolation on lattices providing that the transitions a are continuous and b obey the conventional finite size scaling forms finally we study and compare the direct and reverse processes showing that while the reverse process is different from the direct process for finite size systems the two cases become equivalent in the thermodynamic limit of large l

percolation transitions with nonlocal constraint we investigate percolation transitions in a nonlocal network model numerically in this model each node has an exclusive partner and a link is forbidden between two nodes whose r neighbors share any exclusive pair the r neighbor of a node x is defined as a set of at most n r neighbors of x where n is the total number of nodes the parameter r controls the strength of a nonlocal effect the system is found to undergo a percolation transition belonging to the mean field universality class for r 1 2 the system undergoes a peculiar phase transition from a nonpercolating phase to a quasicritical phase where the largest cluster size g scales as g n with 0 74 1 in the marginal case with r 1 2 the model displays a percolation transition that does not belong to the mean field universality class

hamiltonian approach for explosive percolation we present a cluster growth process that provides a clear connection between equilibrium statistical mechanics and an explosive percolation model similar to the one recently proposed by d achlioptas et al science 323 1453 2009 we show that the following two ingredients are sufficient for obtaining an abrupt first order transition in the fraction of the system occupied by the largest cluster i the size of all growing clusters should be kept approximately the same and ii the inclusion of merging bonds i e bonds connecting vertices in different clusters should dominate with respect to the redundant bonds i e bonds connecting vertices in the same cluster moreover in the extreme limit where only merging bonds are present a complete enumeration scheme based on treelike graphs can be used to obtain an exact solution of our model that displays a first order transition finally the presented mechanism can be viewed as a generalization of standard percolation that discloses a family of models with potential application in growth and fragmentation processes of real network systems

representations of the superalgebra f4 and young supertableaux the authors present a method for constructing typical as well as atypical finite dimensional representations of the superalgebra f4 for such a purpose young supertableaux are introduced