modules of finite projective dimension with negative intersection multiplicities let x be the image of x in r the main object of this paper is to study modules of finite length and finite projective dimension over r one consequence of our study will be counterexamples to several conjectures concerning the behavior of intersection multiplicities for modules of finite projective dimension let denote length if s is a noetherian ring m n are finitely generated s modules l m is finite and d pdsm we may define following serre is d zs m n z m n 1 tors m n

discriminant subspace analysis a fukunaga koontz approach the fisher linear discriminant fld is commonly used in pattern recognition it finds a linear subspace that maximally separates class patterns according to the fisher criterion several methods of computing the fld have been proposed in the literature most of which require the calculation of the so called scatter matrices in this paper we bring a fresh perspective to fld via the fukunaga koontz transform fkt we do this by decomposing the whole data space into four subspaces with different discriminabilities as measured by eigenvalue ratios by connecting the eigenvalue ratio with the generalized eigenvalue we show where the fisher criterion is maximally satisfied we prove the relationship between fld and fkt analytically and propose a unified framework to understanding some existing work furthermore we extend our theory to the multiple discriminant analysis mda this is done by transforming the data into intraclass and extraclass spaces followed by maximizing the bhattacharyya distance based on our fkt analysis we identify the discriminant subspaces of mda fkt and propose an efficient algorithm which works even when the scatter matrices are singular or too large to be formed our method is general and may be applied to different pattern recognition problems we validate our method by experimenting on synthetic and real data

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

spatial organization in two species annihilation the spatial structure of reactants in the two species annihilation reaction a b 0 is described in one dimension we investigate the distribution of domain sizes and the distributions of nearest neighbor distances between particles of the same and of opposite species the latter two quantities are characterized by a new length scale which is intermediate to the domain size t1 2 and the typical interparticle spacing t1 4 a scaling argument suggests that the typical distance between particles of opposite species or equivalently the gaps between domains grows ast with 3 8 and 1 3 respectively in spatial dimensiond 1 and 2 the average density profile of a single domain is spatially nonuniform with the density decaying to zero linearly as the domain edge is approached this behavior permits a determination of the distribution of nearest neighbor distances of same species reactants the corresponding moments of this distribution exhibit multiscaling which involves geometric averages of different powers of the domain size and the interparticle spacing

wiretap channel with causal state information and secure rate limited feedback in this paper we consider the secrecy capacity of a wiretap channel in the presence of causal state information and secure rate limited feedback in this scenario the causal state information from the channel is available to both the legitimate transmitter and the legitimate receiver in addition the legitimate receiver can send secure feedback to the transmitter at a limited rate r f we derive upper and lower bounds on the secrecy capacity and show that when the channel to the eavesdropper is degraded the bounds are tight and the secrecy capacity is completely characterized the capacity achieving scheme is based on wyner csiszar and korner wiretap coding and two steps of shared key generation one from the state information and one via the noiseless feedback the upper bound is more involved and requires a nontrivial recursive lemma extending previous results in the literature to include both state and feedback we conclude the paper by showing that a few interesting known results can be seen as special cases of the above as well as discussing the case where the source of local randomness at the encoder is limited

f coherent rings with applications to tight closure theory abstract the aim of this paper is to introduce a new class of noetherian rings of prime characteristic via perfect closure and study their basic properties if the perfect closure of a noetherian ring is coherent we call it an f coherent ring some applications are given to tight closure theory in particular we discuss some relationship between f coherent rings and f pure f regular and f injective rings as a main tool we use techniques from valuation theory the final section discusses how the coherent property effects the behavior of tight closure of finitely generated ideals on general perfect rings

history and generality of the cs decomposition abstract it is almost a quarter of a century since chandler davis and william kahan brought together the key ideas of what stewart later completed and defined to be the cs decomposition csd of a partitioned unitary matrix this paper outlines some germane points in the history of the csd pointing out the contributions of jordan of davis and kahan and of stewart and the relationship of the csd to the direct rotation of davis and kato the paper provides an easy to memorize constructive proof of the csd reviews one of its important uses and suggests a motivation for the csd which emphasizes how generally useful it is it shows the relation between the csd and generalized singular value decompositions and points out some useful nullity properties one form of the csd trivially reveals finally it shows how via the qr factorization the csd can be used to obtain interesting results for partitioned nonsingular matrices we suggest the csd be taught in its most general form with no restrictions on the two by two partition and initially with no mention of angles between subspaces

perturbation analysis for the generalized singular value problem this paper discusses perturbation bounds for generalized singular values and for subspaces associated with the generalized singular value decomposition of two matrices having the same number of columns by a limiting procedure from our results we can derive those for the ordinary singular value problem which were proved by l mirsky quart j math oxford 11 1960 pp 50 59 and p a wedin bit 12 1972 pp 99 111

comparison of symbolic and ordinary powers of ideals all given rings in this paper are commutative associative with identity and noetherian recently l ein r lazarsfeld and k smith els discovered a remarkable and surprising fact about the behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0 if h is the largest height of an associated prime of i then i hn i n for all n 0 here if w is the complement of the union of the associated primes of i i t denotes the contraction of i t rw to r where rw is the localization of r at the multiplicative system w their proof depends on the theory of multiplier ideals including an asymptotic version and in particular requires resolution of singularities as well as vanishing theorems we want to acknowledge that without their generosity and quickness in sharing their research this manuscript would not exist our objective here is to give stronger results that can be proved by methods that are in some ways more elementary our results are valid in both equal characteristic 0 and in positive prime characteristic p but depend on reduction to characteristic p we use tight closure methods and in consequence we need neither resolution of singularities nor vanishing theorems that may fail in positive characteristic for the most basic form of the result all that we need from tight closure theory is the definition of tight closure and the fact that in a regular ring every ideal is tightly closed we note that the main argument here is closely related to a proof given in hu 5 14 16 p 45 that regular local rings in characteristic p are ufds

canonical elements in local cohomology modules and the direct summand conjecture one of the objectives of this paper is to show that the usual homological consequences of the existence of big cohen macaulay henceforth c m modules e g the new intersection conjecture of peskine szpiro and roberts and the evans griffith syzygy conjecture follow from the direct summand conjecture when the residual characteristic of the local ring is positive this gives a new and substantially more elementary proof of the standard homological conjectures in case the characteristic of the ring itself is positive and reduces the general case of all these conjectures to one rather down to earth conjecture of course this places the direct summand conjecture in a position of central importance so that it now merits an allout attack some partial results on this problem are given in section 6 the conjecture asserts that a regular noetherian ring r is a direct summand as an r module of every module finite extension ring s 3 r the other main objective of this paper is to formulate and develop a theory of certain canonical elements in the local cohomology of special modules of syzygies neither the modules of syzygies nor the induced maps between them are canonical but the identification between the canonical elements is independent of the choices in particular a canonical element qr e hz syz k is associated see section 3 for details with each ndimensional local ring r m k rings are commutative associative with identity local ring means noetherian ring with a unique maximal ideal the conjecture that qr is nonzero for all local rings r turns out to be equivalent to the direct summand conjecture for a given r an infinite family of cases of the latter implies the former

big cohen macaulay algebras and seeds in this article we delve into the properties possessed by algebras which we have termed seeds that map to big cohen macaulay algebras we will show that over a complete local domain of positive characteristic any two big cohen macaulay algebras map to a common big cohen macaulay algebra we will also strengthen hochster and huneke s weakly functorial existence result for big cohen macaulay algebras by showing that the seed property is stable under base change between complete local domains of positive characteristic we also show that every seed over a positive characteristic ring maps to a balanced big cohen macaulay algebra that is an absolutely integrally closed adically separated quasilocal domain

quasilength latent regular sequences and content of local cohomology abstract the quasilength of a finitely generated module that is killed by a power of a finitely generated ideal i is introduced it is the length of a shortest filtration of the module with factors that are cyclic modules killed by i this notion is then used to define a notion of content for the dth local cohomology module of a ring or module with support in an ideal generated by d elements in the case of a ring which is central this content is a real number between 0 and 1 it is not known whether it is independent of the choice of the d generators nor whether it can change if the generators are replaced by powers in positive prime characteristic it is shown that the content is always 0 or 1 this is an open question in equal characteristic 0 and in mixed characteristic it is conjectured that if the elements form a system of parameters in a local ring of dimension d then the content is 1 this is proved if the ring contains a field in mixed characteristic it is an open question in dimension 3 or more and implies the direct summand conjecture the relationship between the notions of quasilength and content and the property that a sequence of elements can become a regular sequence on a ring or module after base change is explored

perturbation analysis of the canonical subspaces abstract this paper presents a perturbation analysis of the canonical subspaces of a matrix pair a b it is shown that if the perturbations in the matrices a b do not change the ranks of each matrix then the perturbation bounds of the canonical subspaces depend upon the perturbations and the scaled condition numbers of the matrix pair as well as upon the gap of the corresponding canonical correlations

existence of almost cohen macaulay algebras implies the existence of big cohen macaulay algebras abstract in 1 the dagger closure is extended over finitely generated modules over noetherian local domain r m and it is proved to be a dietz closure in this short note we show that it also satisfies the algebra axiom of 9 and this leads to the following result of this paper for a complete noetherian local domain if it is contained in an almost cohen macaulay domain then there exists a balanced big cohen macaulay algebra over it

secure transmission with multiple antennas part ii the mimome wiretap channel the capacity of the gaussian wiretap channel model is analyzed when there are multiple antennas at the sender intended receiver and eavesdropper the associated channel matrices are fixed and known to all the terminals a computable characterization of the secrecy capacity is established as the saddle point solution to a minimax problem the converse is based on a sato type argument used in other broadcast settings and the coding theorem is based on gaussian wiretap codebooks at high signal to noise ratio snr the secrecy capacity is shown to be attained by simultaneously diagonalizing the channel matrices via the generalized singular value decomposition and independently coding across the resulting parallel channels the associated capacity is expressed in terms of the corresponding generalized singular values it is shown that a semi blind masked multi input multi output mimo transmission strategy that sends information along directions in which there is gain to the intended receiver and synthetic noise along directions in which there is not can be arbitrarily far from capacity in this regime necessary and sufficient conditions for the secrecy capacity to be zero are provided which simplify in the limit of many antennas when the entries of the channel matrices are independent and identically distributed the resulting scaling laws establish that to prevent secure communication the eavesdropper needs three times as many antennas as the sender and intended receiver have jointly and that the optimum division of antennas between sender and intended receiver is in the ratio of 2 1

fractal growth by local sensitive epidemic model lsem simulating the internal feeding systems ifs abstract the fractal systems may be classified into two families which are distinguishable by a different nutrient feeding during their growth process for the internal feeding systems ifs the source is directly connected to the growing interface while for the external feeding systems efs it is placed far away so that the released particles have to execute random walks before adhering to the aggregate a generalized model nicknamed local sensitive epidemic model lsem is developed to unify the main models simulating the growth of the ifsystems by combining all their growth parameters gp and growth rules gr this model considers in addition i a particle interaction extended to the kth neighbour which may be attractive or repulsive ii a continuous updating of the local characteristics of the surrounding medium and iii an aggregate relaxation which tends to optimize the total energy during or after the growth process the coupled effects of all these parameters allow one to simulate the main fractal phases observed experimentally and to discuss the conditions under which they may be obtained as well as their different morphological transitions

a tree of generalizations of the ordinary singular value decomposition abstract it is shown how to generalize the ordinary singular value decomposition of a matrix into a combined factorization of any number of matrices we propose to call these factorizations generalized singular value decompositions for two matrices this reduces to the product and quotient singular value decompositions one of the factorizations for three matrices is the restricted singular value decomposition these generalizations form a tree of factorizations where at level k for k matrices there are 2 k factorizations not all of which are independent the different levels are related to each other in a recursive fashion any generalized singular value decomposition for k matrices can be constructed from a decomposition for k 1 matrices this results in an inductive proof which uses only the ordinary singular value decomposition several examples are analysed in detail

spatial structure in diffusion limited two particle reactions we analyze the limiting behavior of the densities a t and b t and the random spatial structure r a t b t for the diffusion controlled chemical reaction a b inert for equal initial densities b 0 b 0 there is a change in behavior fromd 4 where a t b t c td 4 tod 4 where a t b t c t ast the termc depends on the initial densities and changes withd there is a corresponding change in the spatial structure ind 4 both particle types are after large times present locally in concentrations not depending on type or location ind 4 both particle types are present locally but with random concentrations and the process tends to a limit

class and rank of differential modules a differential module is a module equipped with a square zero endomorphism this structure underpins complexes of modules over rings as well as differential graded modules over graded rings we establish lower bounds on the class a substitute for the length of a free complex and on the rank of a differential module in terms of invariants of its homology these results specialize to basic theorems in commutative algebra and algebraic topology one instance is a common generalization of the equicharacteristic case of the new intersection theorem of hochster peskine p roberts and szpiro concerning complexes over commutative noetherian rings and of a theorem of g carlsson on differential graded modules over graded polynomial rings

complex dynamics in initially separated reaction diffusion systems we review recent developments in the study of the diffusion reaction systems of the type a b c in which the reactants are initially separated we consider the case where the a and b particles are initially placed uniformly in euclidean space at x 0 and x 0 respectively we find that whereas for d 2 the mean field exponent characterizes the width of the reaction zone fluctuations are relevant in the one dimensional system we present analytical and numerical results for the reaction rate on fractals and percolation systems at criticality we also study the case where the particles are levy flights in d 1 finally we consider experimentally analytically and numerically the reaction a bstatic c where species a diffuses from a localized source

dimension and singularity theory for local rings of finite embedding dimension abstract in this paper we develop an algebraic theory for local rings of finite embedding dimension several extensions of krull dimension are proposed which are then used to generalize singularity notions from commutative algebra finally variants of the homological theorems are shown to hold in equal characteristic this theory is then applied to noetherian local rings in order to get i over a cohen macaulay local ring uniform bounds on the betti numbers of a cohen macaulay module in terms of dimension and multiplicity and similar bounds for the bass numbers of a finitely generated module ii a characterization for being respectively analytically unramified analytically irreducible unmixed quasi unmixed normal cohen macaulay pseudo rational or weakly f regular in terms of certain uniform arithmetic behavior iii in mixed characteristic the improved new intersection theorem when the residual characteristic or ramification index is large with respect to dimension and some other numerical invariants

multiple input multiple output gaussian broadcast channels with confidential messages this paper considers the problem of secret communication over a two receiver multiple input multiple output mimo gaussian broadcast channel the transmitter has two independent messages each of which is intended for one of the receivers but needs to be kept asymptotically perfectly secret from the other it is shown that surprisingly under a matrix power constraint both messages can be simultaneously transmitted at their respective maximal secrecy rates to prove this result the mimo gaussian wiretap channel is revisited and a new characterization of its secrecy capacity is provided via a new coding scheme that uses artificial noise an additive prefix channel and random binning

closure operations that induce big cohen macaulay modules and classification of singularities abstract geoffrey dietz introduced a set of axioms for a closure operation on a complete local domain r so that the existence of such a closure operation is equivalent to the existence of a big cohen macaulay module these closure operations are called dietz closures in complete rings of characteristic p 0 tight closure and plus closure satisfy the axioms we define module closures and discuss their properties for many of these properties there is a smallest closure operation satisfying the property in particular we discuss properties of big cohen macaulay module closures and prove that every dietz closure is contained in a big cohen macaulay module closure using this result we show that under mild conditions a ring r is regular if and only if all dietz closures on r are trivial finally we show that solid closure in equal characteristic 0 integral closure and regular closure are not dietz closures and that all dietz closures are contained in liftable integral closure

how to rescue solid closure abstract we define a closure operation for ideals in a commutative ring which has all the good properties of solid closure at least in the case of equal characteristic but such that also every ideal in a regular ring is closed this gives in particular a kind of tight closure theory in characteristic zero without referring to positive characteristic

on the canonical element conjecture the canonical element conjecture is proved in the following two cases i depth a dim a 1 h a is decomposable ii depth a dim a 1 hm a v is cyclic the equivalence of the c e c and the improved new intersection theorem is also established in h m hochster has assigned a canonical element qa e h syz k with each n dimensional local ring a m k i e commutative noetherian local ring a with a unique maximal ideal m and k a m and has conjectured that 71a 0 for every such a in elementary terms this conjecture means the following h for every free resolution f a8i asi i a k 0 of k and for every system of parameters x1 x of a if 4 is any map of complexes k x a f which lifts the quotient surection a xl x k then 4 kn x a asn is nonzero hochster has shown that the above conjecture henceforth c e c is equivalent to the direct summand conjecture and implies the improved new intersection conjecture which can be stated as follows improved new intersection conjecture let a m be a local ring and f be a finite free complex of finitely generated free modules o fd fo such that hi f has finite length for i 0 and ho f has a nonzero minimal generator v such that avp has finite length then dim r d it is well known that the above conjecture henceforth i n i c implies several other homological conjectures in 1 of this paper we prove that a satisfies c e c a satisfies i n i c 1 4 in proving this we come across in 1 3 another equivalent version of c e c which shows that actually c e c boils down to understanding the relations among the resolutions of homologies of k x a and itself in 2 we study the case when depth a dim a 1 we prove 2 2 that a satisfies c e c in the following cases i when h j1 a v is decomposable this implies hi x a is decomposable ii when h 71 a v is cyclic this implies hi x a v is cyclic received by the editors november 18 1985 1980 mathematics subject classification 1985 revision primary 13h10 the author was supported in part by grants from the national science foundation 1987 american mathematical society 0002 9947 87 1 00 25 per page

lower bounds on projective levels of complexes abstract for an associative ring r the projective level of a complex f is the smallest number of mapping cones needed to build f from projective r modules we establish lower bounds for the projective level of f in terms of the vanishing of homology of f we then use these bounds to derive a new version of the new intersection theorem for level when r is a commutative noetherian local ring

asymptotics of reaction diffusion fronts with one static and one diffusing reactant abstract the long time behavior of a reaction diffusion front between one static e g porous solid reactant a and one initially separated diffusing reactant b is analyzed for the mean field reaction rate density r a b k a m b n a uniformly valid asymptotic approximation is constructed from matched self similar solutions in a reaction front of width w t where r t enters the dominant balance and a diffusion layer of width w t 1 2 where r is negligible the limiting solution exists if and only if m n 1 in which case the scaling exponents are uniquely given by m 1 2 m 1 and m m 1 in the diffusion layer the common ad hoc approximation of neglecting reactions is given mathematical justification and the exact transient decay of the reaction rate is derived the physical effects of higher order kinetics m n 1 such as the broadening of the reaction front and the slowing of transients are also discussed

extended plus closure and colon capturing abstract three results concerning the colon capturing property for extended plus closures in excellent mixed characteristic rings are demonstrated it is shown that the extended plus closure has the colon capturing property for arbitrary sets of three parameters it is also seen that establishing the colon capturing property for the extended plus closure is sufficient to guarantee the existence of balanced big cohen macaulay algebras also for three dimensional complete domains the second local cohomology of r is annihilated by arbitrarily small powers of every non unit

mimo multiple access channel with an arbitrarily varying eavesdropper secrecy degrees of freedom a two transmitter gaussian multiple access wiretap channel with multiple antennas at each of the nodes is investigated the channel matrices of the legitimate users are fixed and revealed to all the terminals whereas the channel matrices of the eavesdropper are arbitrarily varying and only known to the eavesdropper the secrecy degrees of freedom s d o f region under a strong secrecy constraint is characterized a transmission scheme that orthogonalizes the transmit signals of the two users at the intended receiver and uses a single user wiretap code for each user is shown to achieve the s d o f region the converse involves establishing an upper bound on a weighted sum rate expression this is accomplished by using induction where at each step one combines the secrecy and multiple access constraints associated with an adversary eavesdropping a carefully selected group of sub channels

fontaine rings and local cohomology abstract we use a construction due to fontaine to construct rings with almost vanishing local cohomology from rings of mixed characteristic and discuss the question of using this method to construct almost cohen macaulay algebras over the original ring we also show that the existence of almost cohen macaulay algebras implies the monomial conjecture and give an example to show how this procedure can be carried out in a nontrivial case

axiomatic closure operations phantom extensions and solidity abstract in this article we generalize a previously defined set of axioms for a closure operation that induces balanced big cohen macaulay modules while the original axioms were only defined in terms of finitely generated modules these new ones will apply to all modules over a local domain the new axioms will lead to a notion of phantom extensions for general modules and we will prove that all modules that are phantom extensions can be modified into balanced big cohen macaulay modules and are also solid modules as a corollary if r has characteristic p 0 and is f finite then all solid algebras are phantom extensions if r also has a big test element e g if r is complete then solid algebras can be modified into balanced big cohen macaulay modules hochster and huneke have previously demonstrated that there exist solid algebras that cannot be modified into balanced big cohen macaulay algebras we also point out that tight closure over local domains in characteristic p generally satisfies the new axioms and that the existence of a big cohen macaulay module induces a closure operation satisfying the new axioms

a continuous jacobi like approach to the simultaneous reduction of real matrices abstract the problem of simultaneous reduction of real matrices by either orthogonal similarity or orthogonal equivalence transformations is considered based on the jacobi idea of minimizing the sum of squares of the complementary part of the desired form to which matrices are reduced the projected gradient method is used in this paper it is shown that the projected gradient of the objective function can be formulated explicitly this gives rise to a system of ordinary differential equations that can be readily solved by numerical software the advantages of this approach are that the desired form to which matrices are reduced can be almost arbitrary and that if a desired form is not attainable then the limit point of the corresponding differential equation gives a way of measuring the distance from the best reduced matrices to the nearest matrices that have the desired form the general procedure for deriving these differential equations is discussed the framework can be generalized to the complex valued case some applications are given

universal secure network coding via rank metric codes the problem of securing a network coding communication system against an eavesdropper is considered the network implements linear network coding to deliver n packets from source to each receiver and the adversary can eavesdrop on arbitrarily chosen links the objective is to provide reliable communication to all receivers while guaranteeing that the source information remains information theoretically secure from the adversary a coding scheme is proposed that can achieve the maximum possible rate of n packets the scheme which is based on rank metric codes has the distinctive property of being universal it can be applied on top of any communication network without requiring knowledge of or any modifications on the underlying linear network code the only requirement of the scheme is that the packet length be at least n which is shown to be strictly necessary for universal communication at the maximum rate a further scenario is considered where the adversary is allowed not only to eavesdrop but also to inject up to t erroneous packets into the network and the network may suffer from a rank deficiency of at most in this case the proposed scheme can be extended to achieve the rate of n 2t packets this rate is shown to be optimal under the assumption of zero error communication

mimo broadcast channel with an unknown eavesdropper secrecy degrees of freedom we study a multi antenna broadcast channel with two legitimate receivers and an external eavesdropper we assume that the channel matrix of the eavesdropper is unknown to the legitimate terminals but satisfies a maximum rank constraint as our main result we characterize the associated secrecy degrees of freedom for the broadcast channel with common and private messages we show that a direct extension of the single user wiretap codebook does not achieve the secrecy degrees of freedom our proposed optimal scheme involves decomposing the signal space into a common subspace which can be observed by both receivers and private subspaces which can be observed by only one of the receivers and carefully transmitting a subset of messages in each subspace we also consider the case when each user s private message must additionally remain confidential from the other legitimate receiver and characterize the s d o f region in this case

secrecy in mimo networks with no eavesdropper csit we consider two fundamental multi user channel models the multiple input multiple output mimo wiretap channel with one helper wth and the mimo multiple access wiretap mac wt channel in each case the eavesdropper has k antennas while the remaining terminals have n antennas each we consider a fast fading channel where the channel state information csi of the legitimate receiver is available at the transmitters but no csi at the transmitters csit is available for the eavesdropper s channel we determine the optimal sum secure degrees of freedom s d o f for each channel model for the regime k leq n and show that in this regime the mac wt channel reduces to the wth in the absence of eavesdropper csit for the regime n leq k leq 2n we obtain the optimal linear s d o f and show that the mac wt channel and the wth have the same optimal s d o f when restricted to linear encoding strategies in the absence of any such restrictions we provide an upper bound for the sum s d o f of the mac wt channel in the regime n leq k leq 2n our results show that unlike in the single input single output case there is loss of s d o f for even the wth due to lack of eavesdropper csit when k geq n

the secrecy capacity region of the gaussian mimo multi receiver wiretap channel in this paper we consider the gaussian multiple input multiple output mimo multi receiver wiretap channel in which a transmitter wants to have confidential communication with an arbitrary number of users in the presence of an external eavesdropper we derive the secrecy capacity region of this channel for the most general case we first show that even for the single input single output siso case existing converse techniques for the gaussian scalar broadcast channel cannot be extended to this secrecy context to emphasize the need for a new proof technique our new proof technique makes use of the relationships between the minimum mean square error and the mutual information and equivalently the relationships between the fisher information and the differential entropy using the intuition gained from the converse proof of the siso channel we first prove the secrecy capacity region of the degraded mimo channel in which all receivers have the same number of antennas and the noise covariance matrices can be arranged according to a positive semi definite order we then generalize this result to the aligned case in which all receivers have the same number of antennas however there is no order among the noise covariance matrices we accomplish this task by using the channel enhancement technique finally we find the secrecy capacity region of the general mimo channel by using some limiting arguments on the secrecy capacity region of the aligned mimo channel we show that the capacity achieving coding scheme is a variant of dirty paper coding with gaussian signals