a construction of f 1 as automorphisms of a 196 883 dimensional algebra abstract in this note i announce the construction of the finite simple group f1 whose existence was predicted independently in 1973 by bernd fischer and by me the group has order 246320597611213317 19 23 29 31 41 47 59 71 808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000 and is realized as a group of automorphisms of a 196 883 dimensional commutative nonassociative algebra over the rational numbers which has an associative form equivalently it is a group of automorphisms of a cubic form in 196 883 variables it turns out that all the relevant arguments and calculations may be done by hand furthermore existence of the group f1 implies the existence of a number of other sporadic simple groups for which existence proofs formerly depended on work with computers we are beginning to look upon this group as a friendly giant

strong duality and kkt conditions in nonconvex optimization with a single equality constraint and geometric constraint some topological and geometric characterizations of strong duality for a non convex optimization problem under a single equality and geometric constraints are established in particular a hidden convexity of the conic hull of joint range of the pair of functions associated to the original problem is obtained applications to derive a characterization of the validity of kkt conditions without standard constraints qualification are also discussed it goes beyond the exact penalization technique several examples showing our results provide much more information than those appearing elsewhere are given finally the standard quadratic problem involving a non necessarily polyhedral cone is analyzed in detail

exact penalization of mathematical programs with equilibrium constraints we study theoretical and computational aspects of an exact penalization approach to mathematical programs with equilibrium constraints mpecs in the first part we prove that a mangasarian fromovitz type condition ensures the existence of a stable local error bound at the root of a real valued nonnegative piecewise smooth function a specification to nonsmooth formulations of equilibrium constraints e g complementarity conditions or normal equations provides conditions which guarantee the existence of a nonsmooth exact penalty function for mpecs in the second part we study a trust region minimization method for a class of composite nonsmooth functions which comprises exact penalty functions arising from mpecs we prove a global convergence result for the general method and incorporate a penalty update rule a further specification results in an sqp trust region method for mpecs based on an ell 1 penalty function

the capacity of the semideterministic relay channel corresp the capacity of the class of relay channels with sender x 1 a relay sender x 2 a relay receiver y 1 f x 1 x 2 and ultimate receiver y is proved to be c max min p x 1 x 2 i x 1 x 2 y h y 1 x 2 i x 1 y x 2 y 1

categories premodulaires modularisations et invariants des varietes de dimension 3 an abelian k linear semisimple category having a finite number of simple objects and endowed with a ribbon structure is called premodular it is modular in the sense of turaev if the so called s matrix is invertible a modular category defines invariants of 3 manifolds and a tqft t when is it possible to construct a modularisation of a given premodular category i e a functor to a modular category preserving the structures and dominant in a certain sense it turns out 2 3 that this amounts essentially to making transparent objects trivial we give a full answer to this problem in the case when k is a field of char 0 as well as partial answers in char p under a few obvious hypotheses a premodular category admits a modularisation which is unique th 3 1 and cor 3 5 in char 0 the proof relies on two main ingredients a new and very simple criterion for the s matrix to be invertible 1 1 and deligne s internal characterization of tannakian categories in char 0 d when simple transparent objects are invertible the criterion is simpler 4 2 and the modularisation can be described more explicitly prop 4 4 we conclude with two examples the premodular categories associated with quantum hbox sf sl n and hbox sf pgl n at roots of unity in the first case we obtain modular categories which were built independently by c blanchet b in the second case we obtain modularizations in all the cases where y yokota y found reshetikhin turaev invariants of 3 manifolds thereby improving as well as explaining yokota s results

virasoro frames and their stabilizers for the e8 lattice type vertex operator algebra the concept of a framed vertex operator algebra fvoa is new cf dgh this article contributes to this theory with a full analysis of all virasoro frame stabilizers in v the important example of the e8 level 1 affine kac moody voa which is isomorphic to the lattice voa for the root lattice of e8 c we analyze the frame stabilizers both as abstract groups and as subgroups of aut v e8 c each frame stabilizer is a finite group contained in the normalizer of a 2b pure elementary abelian 2 group in aut v but is not usually a maximal finite subgroup of this normalizer in particular we prove that there are exactly five orbits for the action of aut v on the set of virasoro frames thus settling an open question about v in section 5 of dgh the results about the group structure of the frame stabilizers can be stated purely in terms of modular braided tensor categories so this article contributes also to this theory there are two main viewpoints in our analysis the first is the theory of codes lattices markings and the resulting groups of automorphisms the second is the theory of finite subgroups of lie groups we expect our methods to be applicable to the study of other fvoas and their frame stabilizers appendices present aspects of the theory of automorphism groups of voas in particular there is a general result of independent interest on embedding lattices into unimodular lattices so as to respect automorphism groups and definiteness

outage theorems for mimo block fading channels the connection between the average codeword or frame error probability fep of space time codes and the outage probability over general block fading multiple input multiple output mimo channels is established three archetypal problems are considered under general fading distributions in a single framework wherein the receiver has channel state information whereas the transmitter knows a the fading distribution but not the channel realization b the channel realization but must follow a short term per codeword average power constraint and c the channel realization but is constrained only by a long term average power constraint three telescoping sets of space time codes are defined for a given rate and it is shown that average feps arbitrarily close to the respective outage probabilities for each of the three cases a c can be achieved by codes in each set for sufficiently large frame lengths for the smallest set among the three which contains codes with a spectral norm constraint that is stricter than the average or maximum energy constraints commonly assumed firm sphere packing lower bounds on the fep are obtained and consequently strong converse theorems are proved which assert that the respective outage probabilities also represent the best achievable fep in the large frame length limit moreover the set of spectral norm constrained codes are also shown to be large enough to contain universal codes that can communicate reliably over any channel realization for which the mutual information exceeds the information rate of the code

global error bounds with fractional exponents using the partial order induced by a proper weakly lower semicontinuous function on a reflexive banach space x we give a sufficient condition for f to have error bounds with fractional exponents application is given to identify the set of such exponents for quadratic functions

lagrange multipliers and calmness conditions of order p in this paper by assuming that a non lipschitz penalty function is exact new conditions for the existence of lagrange multipliers are established for an inequality and equality constrained continuously differentiable optimization problem this is done by virtue of a first order necessary optimality condition of the penalty problem which is obtained by estimating dini upper directional derivatives of the penalty function in terms of taylor expansions and a farkas lemma relations among the obtained results and some well known constraint qualifications are discussed

the generalized diversity multiplexing tradeoff of the mimo z interference channel the generalized diversity multiplexing tradeoff gdmt of the two user quasi static fading multi input multi output mimo z interference channel z ic is established for the general case with an arbitrary number of antennas at each node under the assumptions of full channel state information at the transmitters csit and a short term average power constraint in the gdmt framework which captures the rate versus reliability tradeoff in the high signal to noise ratio snr regime the direct link snr and cross link interference to noise ratio inr are allowed to be disparate so that their ratios relative to a nominal snr in the decibel scale i e the snr and inr exponents are arbitrary and fixed it is shown that a simple han kobayashi message splitting partial interference decoding scheme that uses only partial csit in which the second transmitter s signal depends only on its cross link channel matrix and the first user s transmit signal does not need any csit can achieve the full csit gdmt of the mimo z ic the gdmt of the mimo z ic under the no csit assumption is also obtained for some range of multiplexing gains the size of this range depends on the numbers of antennas at the four nodes and the snr and inr exponents of the direct and cross links respectively for certain classes of channels including those in which the interfered receiver has more antennas than do the other nodes or when the inr exponent is greater than a certain threshold the gdmt of the mimo z ic under the no csit assumption is also completely characterized

a unified augmented lagrangian approach to duality and exact penalization in this paper the existence of an optimal path and its convergence to the optimal set of a primal problem of minimizing an extended real valued function are established via a generalized augmented lagrangian and corresponding generalized augmented lagrangian problems in which no convexity is imposed on the augmenting function these results further imply a zero duality gap property between the primal problem and the generalized augmented lagrangian dual problem a necessary and sufficient condition for the exact penalty representation in the framework of a generalized augmented lagrangian is obtained in the context of constrained programs we show that generalized augmented lagrangians present a unified approach to several classes of exact penalization results some equivalences among exact penalization results are obtained

characterizing fj and kkt conditions in nonconvex mathematical programming with applications in this paper we analyze the fritz john and karush kuhn tucker kkt conditions for a g teaux differentiable nonconvex optimization problem with inequality constraints and a geometric constraint set the fritz john condition is characterized in terms of an alternative theorem which covers beyond standard situations while characterizations of kkt conditions without assuming constraints qualifications are related to strong duality of a suitable linear approximation of the given problem and the properties of its associated image mapping such characterizations are suitable for dealing with some problems in structural optimization where most of the known constraint qualifications fail in particular several examples are given showing the usefulness and optimality in a certain sense of our results which provide much more information than those including the mordukhovich normal cone or clarke s appearing elsewhere the case with a single inequality constraint is discussed in detail by establishing

the diversity multiplexing tradeoff of the mimo half duplex relay channel the fundamental diversity multiplexing tradeoff of the three node multi input multi output mimo quasi static rayleigh faded half duplex relay channel is characterized for an arbitrary number of antennas at each node and in which opportunistic scheduling or dynamic operation of the relay is allowed i e the relay can switch between receive and transmit modes at a channel dependent time in this most general case the diversity multiplexing tradeoff is characterized as a solution to a simple two variable optimization problem this problem is then solved in closed form for special classes of channels defined by certain restrictions on the numbers of antennas at the three nodes the key mathematical tool developed here that enables the explicit characterization of the diversity multiplexing tradeoff is the joint eigenvalue distribution of three mutually correlated random wishart matrices besides being relevant here this distribution result is interesting in its own right previously without actually characterizing the diversity multiplexing tradeoff the optimality in this tradeoff metric of the dynamic compress and forward protocol based on the classical compress and forward scheme of cover and el gamal was shown by yuksel and erkip however this scheme requires global channel state information at the relay in this paper the so called quantize map and forward qmf coding scheme is adopted as the achievability scheme with the added benefit that it achieves optimal tradeoff with only the knowledge of the channel dependent switching time at the relay node moreover in special classes of the mimo half duplex relay channel the optimal tradeoff is shown to be attainable even without this knowledge such a result was previously known only for the half duplex relay channel with a single antenna at each node also via the qmf scheme more generally the explicit characterization of the tradeoff curve in this study enables the in depth comparisons herein of full duplex versus half duplex relaying as well as static versus dynamic relaying both as a function of the numbers of antennas at the three nodes

a generalized second order derivative in nonsmooth optimization in this work a new notion of generalized second order directional derivative and generalized hessian for nonsmooth real valued functions is studied the general properties of these mathematical objects are investigated together with some calculus rules that may facilitate their practical computation two applications of these derivatives in optimization theory are considered first to obtaining necessary and sufficient second order optimality conditions for problems with or without constraints and second to extending the newton method for the minimization of a mathcal c 1 1 function

diversity multiplexing tradeoff for the interference channel with a relay we study the diversity multiplexing tradeoff dmt for the slow fading interference channel with a relay icr we derive four inner bounds on the dmt region the first is based on the compress and forward cf relaying scheme the second is based on the decode and forward df relaying scheme and the last two bounds are based on the half duplex hd and full duplex fd amplify and forward af schemes for the cf and df schemes we find conditions on the channel parameters and the multiplexing gains under which the corresponding inner bound achieves the optimal dmt region we also identify the cases in which the dmt region of the icr corresponds to that of two parallel slow fading relay channels implying that interference does not decrease the dmt for each pair and that a single relay can be dmt optimal for two pairs simultaneously for the hd af scheme we derive conditions on the channel coefficients under which the proposed scheme achieves the optimal dmt for the af based relay channel finally we identify the conditions under which adding a relay strictly enlarges the dmt region relative to the interference channel without a relay

existence of augmented lagrange multipliers reduction to exact penalty functions and localization principle in this article we present new general results on existence of augmented lagrange multipliers we define a penalty function associated with an augmented lagrangian and prove that under a certain growth assumption on the augmenting function an augmented lagrange multiplier exists if and only if this penalty function is exact we also develop a new general approach to the study of augmented lagrange multipliers called the localization principle the localization principle allows one to study the local behaviour of the augmented lagrangian near global optimal solutions of the initial optimization problem in order to prove the existence of augmented lagrange multipliers

diversity multiplexing tradeoff for the mimo static half duplex relay in this paper we investigate the diversity multiplexing tradeoff dmt of the multiple antenna mimo static half duplex relay channel a general expression is derived for the dmt upper bound which can be achieved by a compress and forward protocol at the relay under certain assumptions the dmt expression is given as the solution of a minimization problem in general and an explicit expression is found when the relay channel is symmetric in terms of number of antennas i e the source and the destination have n antennas each and the relay has m antennas it is observed that the static half duplex dmt matches the full duplex dmt when the relay has a single antenna and is strictly below the full duplex dmt when the relay has multiple antennas besides the derivation of the upper bound involves a new asymptotic study of spherical integrals that is integrals with respect to the haar measure on the unitary group u n which is a topic of mathematical interest in itself

error bounds for lower semicontinuous functions in normed spaces without the convexity or analyticity assumption we study error bounds for an inequality system defined by a general lower semicontinuous function and establish sufficient necessary conditions on the existence of error bounds in infinite dimensional normed spaces some characterizations for a convex inequality system to possess an error bound in a reflexive banach space are also given as applications in dealing with the hoffman error bound result in normed spaces we give a computable lipschitz bound constant which is better than previous lipschitz bound constants in some examples we also consider error bounds for quadratic functions on rn

calmness and error bounds for convex constraint systems in the paper we first compare the concepts of calmness local error bound and locally linear regularity second we present some dual sufficient conditions for local error bound and local linear regularity by using a characterization of the locally linear regularity for a collection of finite nonempty closed and convex subsets we prove that the sufficient conditions provided in r henrion and j outrata j math anal appl 258 2001 pp 110 130 r henrion and a jourani siam j optim 13 2002 pp 520 534 r henrion a jourani and j outrata siam j optim 13 2002 pp 603 618 imply not only the calmness but also the existence of local error bounds for the convex constraint system in an infinite dimensional setting

optimality conditions via exact penalty functions in this paper we study kkt optimality conditions for constrained nonlinear programming problems and strong and mordukhovich stationarities for mathematical programs with complementarity constraints using l p penalty functions with 0 leq p leq1 we introduce some optimality indication sets by using contingent derivatives of penalty function terms some characterizations of optimality indication sets are obtained by virtue of the original problem data we show that the kkt optimality condition holds at a feasible point if this point is a local minimizer of some l p penalty function with p belonging to the optimality indication set our result on constrained nonlinear programming includes some existing results from the literature as special cases

smoothing methods for nonsmooth nonconvex minimization we consider a class of smoothing methods for minimization problems where the feasible set is convex but the objective function is not convex not differentiable and perhaps not even locally lipschitz at the solutions such optimization problems arise from wide applications including image restoration signal reconstruction variable selection optimal control stochastic equilibrium and spherical approximations in this paper we focus on smoothing methods for solving such optimization problems which use the structure of the minimization problems and composition of smoothing functions for the plus function x many existing optimization algorithms and codes can be used in the inner iteration of the smoothing methods we present properties of the smoothing functions and the gradient consistency of subdifferential associated with a smoothing function moreover we describe how to update the smoothing parameter in the outer iteration of the smoothing methods to guarantee convergence of the smoothing methods to a stationary point of the original minimization problem