capacity of mimo channels with antenna selection this correspondence studies the capacity of multiple input multiple output mimo channels in the presence of antenna selection antenna selection reduces the complexity of the radio devices and requires only a small amount of channel state feedback for high signal to noise ratio snr we define excess rate as the constant term in the expansion of the ergodic capacity in terms of snr it is shown that this value is representative of the channel state information csi at the transmitter the asymptotic behavior of the excess rate is then analyzed for three cases complete csi no csi and partial csi at transmitter antenna selection while water filling provides a excess rate that increases logarithmically in m the number of transmit antennas the excess rate of transmit antenna selection behaves only like log log m for the low snr case we use the concept of channel gain a measure introduced by verdu we show that channel gain for antenna selection increases only logarithmically in m as opposed to water filling channel gain which increases linearly in m the same techniques are also applied to the receive selection and corresponding results are noted in high and low snr regimes the methodology developed in this correspondence although motivated by antenna selection is fairly general and can be used for any system where partial csi is available at the mimo transmitter

the faces of the unit balls of c norms and c spectral norms abstract we give a complete description of the faces of the closed unit ball of an arbitrary c norm on r n and we determine the barycenter the volume and the symmetries of each face as a consequence of a previous result we characterize the faces of the closed unit ball of the corresponding c spectral norm on the space of n m real or complex matrices

isometries for the induced c norm on square matrices and some related results abstract let fn be the linear space of column vectors with n coordinates over f r or c denote by gp n the group of n n generalized permutation matrices i e matrices with exactly one nonzero entry with magnitude 1 in each row and column given a nonzero vector c c1 cn t fn define the c norm of x x1 xn t fn by x c max c t p x p gp n and the induced c norm of a fn n by a c max a y c y c 1 we characterize the isometries for induced c norms to achieve our goal we study the geometric properties of fn n and obtain some inequalities related to the induced c norm as by products these results are of independent interest

l subinvariant norms abstract let l be a compact semigroup of operators on f n where f r or f c a norm p on f n is said to be l subinvariant if p s x p x for all s l x f n if l is a closed subgroup of unitary operators on f n the concept of l subinvariant norm coincides with the concept of l invariant norm studied in c k li n k tsing linear algebra appl 150 1991 179 194 in this paper some basic properties of l subinvariant norms are studied and several results of li and tsing c k li n k tsing linear algebra appl 150 1991 179 194 on group invariant norms are extended to semigroup subinvariant norms as an application necessary and sufficient conditions on a pair of vectors x y f n are obtained such that p x p y holds for each weakly monotonic norm p

the isometries of certain maximum norms abstract let a 1 a m be nonsingular n n matrices over f r or c and let 1 p j for j 1 m we characterize the linear isometries for the norm n on f n defined by n x max l p j a j x 1 j m where l p y denotes the l p norm of y f n as an application we study the conditions on matrices a 1 a m for which n is absolute or a symmetric gauge function

on the extremal structure of least upper bound norms and their dual abstract given finite dimensional real or complex banach spaces e and f with norms e r and f r we denote by n the least upper bound norm induced on l e f some results are given on the extremal structures of b the unit ball of n of its polar b and of b which is the polar of the unit ball of the least upper bound norm n the exposed faces the extreme points and a large family of other faces of b and b are presented it turns out that b is a subset of b the set of tangency points of the surfaces of b and b is completely determined and represented as the union of the exposed faces of b which are normal to rank one mappings we determine sharp bounds on the ranks of mappings in these exposed faces

combining beamforming and space time coding using noisy quantized feedback the goal of combining beamforming and spacetime coding is to obtain full diversity order and to provide additional received power array gain compared to conventional space time codes in this work a class of code constellations is proposed called generalized partly orthogonal designs pods and both high rate and low rate feedback information is incorporated with possible feedback errors a binary symmetric channel bsc model characterizes feedback errors two cases are studied first when the bsc bit error probability is known a priori to the transmission ends and second when it is not known exactly based on a minimum pairwise error probability pep design criterion we design a channel optimized vector quantizer covq for feedback information and a precoder matrix codebook to adjust the transmission codewords the attractive property of our combining scheme is that it converges to conventional space time coding with low rate and erroneous feedback and to directional beamforming with high rate and error free feedback this scheme also shows desirable robustness against feedback channel modeling mismatch

linear preserver problems a brief introduction and some special techniques abstract linear preserver problems concern the characterization of linear operators on matrix spaces that leave certain functions subsets relations etc invariant the earliest papers on linear preserver problems date back to 1897 and a great deal of effort has been devoted to the study of this type of question since then we present a brief picture of the subject aiming at giving a gentle introduction to the reader then we describe some techniques used in our recent papers on this type of problem

certain isometries on rn abstract let r n be the linear space of all real column vectors with n coordinates given x x 1 x n t r n its p k norm is defined by x p k max x i 1 p x i k p 1 p 1 i 1 k n where k is a positive integer satisfying 1 k n and p satisfies 1 p whereas its c norm is defined by x c max c t px p is a generalized permutation matrix for any nonzero c r n this paper characterizes the isometries for c norms and p k norms some related results are also considered

lieb functions and m positivity of norms abstract some properties of 2 or 3 positive maps on matrix spaces are proved where special attention is given to norms it is shown that for a locally bounded function the definition of a lieb function can be simplified somewhat unitarily invariant norms are characterized by a certain monotonicity property the set of 2 positive numerical radii is described and an example of a 3 positive norm is obtained

a limited feedback joint precoding for amplify and forward relaying this paper deals with the practical precoding design for a dual hop downlink with multiple input multiple output mimo amplify and forward relaying first assuming that full channel state information csi of the two hop channels is available a suboptimal dual hop joint precoding scheme i e precoding at both the base station and relay station is investigated based on its structure a scheme of limited feedback joint precoding using joint codebooks is then proposed which uses a distributed codeword selection to concurrently choose two joint precoders such that the feedback delay is considerably decreased finally the joint codebook design for the limited feedback joint precoding system is analyzed and results reveal that independent codebook designs at the base station and relay station using the conventional grassmannian subspace packing method is able to guarantee that the overall performance of the dual hop joint precoding scheme improves with the size of each of the two codebooks simulation results show that the proposed dual hop joint precoding system using distributed codeword selection scheme exhibits a rate or ber performance close to the one using the optimal centralized codeword selection scheme while having lower computational complexity and shorter feedback delay

operator norms multiplicativity factors and c numerical radii abstract let v be a normed vector space over c let b v denote the algebra of linear bounded operators on v and let n be an arbitrary seminorm or norm on b v in this paper we discuss multiplicativity factors for n i e constants 0 for which n n is submultiplicative we find that while in the finite dimensional case nontrivial indefinite seminorms have no multiplicativity factors and norms do have multiplicativity factors in the infinite dimensional case n may or may not have such factors our results are then applied in order to compute multiplicativity factors for certain generalizations of the classical numerical radius called c numerical radii this is done with the help of a combinatorial inequality which seems to be of independent interest

duality between some linear preserver problems ii isometries with respect to c special norms and matrices with fixed singular values let fm n m n denote the linear space of all m n complex or real matrices according as f c or r let c c1 cm 0 be such that c1 cm 0 the c spectral norm of a matrix a fm n is the quantity a c i imci i a where 1 a m a are the singular values of a let d d1 dm 0 where d1 dm 0 we consider the linear isometries between the normed spaces fm n c and fm n d and prove that they are dual transformations of the linear operators which map l d onto l c where l c x fm n x has singular values c1 cm

quantized feedback information in orthogonal space time block coding this work considers how the presence of quantized channel information obtained from a feedback link may be utilized for determining a transmit weighting matrix that improves the performance of a predetermined orthogonal space time block ostb code to reduce the effects of feedback delay quantization errors and feedback channel bit errors methods based on vector quantization for noisy channels are used in the design of the feedback link the resulting transmission scheme and feedback link take the imperfect nature of the channel information into account while combining the benefits of conventional beamforming with those provided by ostb coding

distributed differential space time coding for wireless relay networks distributed space time coding is a cooperative transmission scheme for wireless relay networks with this scheme antennas of the distributive relays work as transmit antennas of the sender and generate a space time code at the receiver it achieves the maximum diversity although the scheme needs no channel information at relays it does require full channel information both the channels from the transmitter to relays and the channels from relays to the receiver at the receiver in this paper we propose a differential transmission scheme which requires channel information at neither relays nor the receiver for wireless relay networks as distributed space time coding can be seen as the counterpart of space time coding in the network setting this scheme is the counterpart of differential space time coding compared to coherent distributed space time coding the differential scheme is 3db worse in addition we show that alamouti square real orthogonal and sp 2 codes can be used differentially in networks with corresponding numbers of relays we also propose distributed differential space time codes that work for networks with any number of relays using circulant matrices

g invariant norms and g c radii abstract let v be a finite dimensional inner product space over f r or c and let g be a closed subgroup of the group of unitary operators on v a norm or a seminorm on v is said to be g invariant if g x x for all g g x v the concept of g invariant norm specializes to many interesting particular cases such as the absolute norms on f n symmetric gauge functions on r n unitarily invariant norms on f m n etc which are of wide research interest in this paper we study the general properties of g invariant norms our main strategy is to study g invariant norms via the g c radius rg c on v defined by r g c x max x g c g g where c v it is shown that the g c radii are very important g invariant seminorms because every g invariant norm or seminorm admits a representation in terms of them as a result one may focus attention on g c radii in order to get results on g invariant norms we study the norm properties of g c radii and obtain various results relating g invariant norms and g c radii the linear operators on v that preserve g invariant norms g invariant seminorms or g c radii are also investigated several open questions are mentioned

faces of the unit ball of a unitarily invariant norm abstract let be a unitarily invariant norm on the space of all n m complex matrices and let g r n r be the symmetric gauge function associated with that is to say we have a g s a for any a where s a is the nonincreasing sequence of singular values of a in this article we consider the relationship between the facial structures of the closed unit balls of g and which we denote by bg and b our main result gives a complete characterization of the faces of b in terms of the faces of bg for with each face e of b we associate a standard face f e of bg i e f e is a face of bg whose barycenter is a nonnegative nonincreasing n vector conversely each standard face f of bg originates a set of n m matrices say the set m f which is proven to be a face of b our main result also asserts that any face e of b is of the form q m fr where q and r are unitary matrices of appropriate orders and f f e we then explore the main result and draw some of its consequences for example we show that the singular values of the barycenter of a face e of b have the same multiplicities as the barycenter of f e formulas are given which relate the dimensions of e and f e the relative interior and the relative boundary of e are characterized in terms of the relative interior and the relative boundary of f e we completely describe the face of b generated by a matrix a our results also cover the case of real matrices

linear maps relating different unitary similarity orbits or different generalized numerical ranges abstract let m be the complex linear space m n of n n complex matrices or the real linear space h n of n n hermitian matrices for c m its unitary similarity orbit is the set u c ucu u unitary and its circular unitary similarity orbit is the set v c x f 1 x u c where f is the scalar field c or r according as m m n or m h n related to u c and v c are the c numerical range and the c numerical radius of a m defined by w c a tr ax x u c and r c a max z z w c a respectively let c d h n we study the linear operators t on m satisfying one of the following properties i w d t a w c a for all a m ii r d t a r c a for all a m iii t u d u c iv t v d v c in particular we determine the conditions on c and d for the existence of a linear operator t on m satisfying any one of the conditions i iv and characterize such an operator if it exists

permutation invariant norms abstract a norm or seminorm on on r n is permutation invariant if px x for all n n permutation matrices p and for all x r n we present a systematic study of permutation invariant seminorms their relations with other types of norms on matrices are discussed in addition we consider a special class of permutation invariant seminorms the c radii defined and denoted by r c x max c t px p a permutation for any given c r n it is shown that r c are the building blocks of all permutation invariant seminorms if the entries of c are not all equal and their sum is not 0 then r c is a norm on r n for such r c we study their norm properties and characterize their isometry groups

combining beamforming and orthogonal space time block coding multiple transmit and receive antennas can be used in wireless systems to achieve high data rate communication efficient space time codes have been developed that utilize a large portion of the available capacity these codes are designed under the assumption that the transmitter has no knowledge about the channel in this work on the other hand we consider the case when the transmitter has partial but not perfect knowledge about the channel and how to improve a predetermined code so that this fact is taken into account a performance criterion is derived for a frequency nonselective fading channel and then utilized to optimize a linear transformation of the predetermined code the resulting optimization problem turns out to be convex and can thus be efficiently solved using standard methods in addition a particularly efficient solution method is developed for the special case of independently fading channel coefficients the proposed transmission scheme combines the benefits of conventional beamforming with those given by orthogonal space time block coding simulation results for a narrow band system with multiple transmit antennas and one or more receive antennas demonstrate significant gains over conventional methods in a scenario with nonperfect channel knowledge

isometries for unitarily invariant norms a survey of linear isometries for unitarily invariant norms on real or complex rectangular matrices is given which includes some latest development on the topic a result on isometries for unitarily invariant norms without the linearity assumption is presented related results and problems are discussed

some aspects of the theory of norms abstract let v be a finite dimensional vector space motivated by theory or applications one might want to consider different kinds of norms on v in this paper we discuss some results and problems involving different classes of norms on a vector space studied by this author in the past few years the paper consists of five sections section 1 concerns the conditions on two vectors x y v satisfying x y for all in a certain class of norms section 2 concerns the isometry groups of g invariant norms i e norms that satisfy g x x for all x v and for all g g where g is a group of unitary orthogonal operators on v section 3 concerns g invariant norms that satisfy some special properties section 4 concerns the best approximation s x 0 t of y where y t v with respect to different kinds of norms additional open problems topics and references are mentioned in section 5

duality between some linear preserver problems iii c spectral norms and skew symmetric matrices with fixed singular values abstract let f denote either the complex field c or the real field r let v be sn f or kn f the vector spaces of all n n symmetric and skew symmetric matrices respectively over f for c c1 cn 0 with c1 cn 0 the c spectral norm of a matrix a v is the quantity a c i l n c i i a where 1 a n a are the singular values of a let d d1 dn 0 with d1 dn 0 we study the linear isometries between the normed spaces v c and v d by using the fact that they are dual transformations of the linear operators which map d onto c where c x v x has singular values c 1 c n it is shown that such isometries and hence their dual transformations exist if and only if c and d are scalar multiples of each other in such case we completely determine the structure of such isometries and prove that they and their dual transformations belong to a same class of operators in the proof we obtain characterizations of the extreme points of the unit ball in v for different cases with respect to c which is of independent interest

the isometries and the g invariance of certain seminorms abstract let f be the field of real or complex numbers and let g be a subgroup of the general linear group gl n f if a is an m n matrix over f let a be the seminorm on f n defined by x a ax for all x f n in this paper we characterize the linear isometries for the seminorm a and study the conditions on a for which a is g invariant that is sx a x a for all x f n and all s g as a special case we describe all matrices a for which a is absolute or a symmetric gauge function

exposed faces and duality for symmetric and unitarily invariant norms abstract let be a unitarily invariant norm on the space of real or complex n m matrices and g the associated symmetric gauge function thus a g s a where s a is the decreasing sequence of singular values of a denote by b and bg the closed unit balls of and g in a previous paper we showed a close relationship between the faces of b and those of bg in particular to each face of b we associated a standard face of bg and we used this association to completely describe the matrices that are members of an individual face of in the present paper we consider the duality operator that transforms each exposed face of b into an exposed face of the unit ball of s dual and the duality operator that does the same with exposed faces of bg we show that these two operators are very nicely related among other results we prove the following given an exposed face e of b the standard face associated with the dual of e is precisely the dual of the standard face of bg associated with e e is an exposed face of b if and only if its associated standard face is an exposed face of bg as a by product we completely determine the subdifferential of in terms of the subdifferential of g and we completely characterize the matrices that are dual to a given matrix a with respect to

linear operators preserving the p q numerical range abstract let bd m be h n or c n n the real linear space of all n n hermitian matrices and the complex linear space of all n n complex matrices respectively for 1 p q n the p q numerical radius of a m is defined and denoted by r p q a e p x ax x d n q x x i q where ep y denotes the pth elementary symmetric function on the eigenvalues of y we characterize the linear operators on m that satisfy rp q a rp q a for all a m

on the structure of limited feedback beamforming codebooks for amplify and forward relay networks we determine necessary conditions on the structure of symbol error rate ser optimal compact quantizer codebooks for limited feedback beamforming in wireless networks with one transmitter receiver pair and r parallel amplify and forward relays we call a codebook small if its cardinality is less than r and large otherwise a d codebook depends on the power constraints and can be optimized accordingly while an i codebook remains fixed it was previously shown that any i codebook that contains the single relay selection srs codebook achieves the full diversity order r we prove the following every full diversity i codebook contains the srs codebook and thus is necessarily large in general as the power constraints grow to infinity the limit of an ser optimal large d codebook contains an srs codebook provided that it exists for small codebooks the maximal diversity is equal to the codebook cardinality every diversity optimal small i codebook is an orthogonal multiple relay selection omrs codebook moreover the limit of an ser optimal small d codebook is an omrs codebook we observe that srs is nothing but a special case of omrs for codebooks with cardinality equal to r as a result we call omrs as the universal necessary condition for codebook optimality finally we confirm our analytical findings through simulations

the fibration method over real function fields let mathbb r c be the function field of a smooth irreducible projective curve over mathbb r let x be a smooth projective geometrically irreducible variety equipped with a dominant morphism f onto a smooth projective rational variety with a smooth generic fibre over mathbb r c assume that the cohomological obstruction introduced by colliot thelene is the only one to the local global principle for rational points for the smooth fibres of f over mathbb r c valued points then we show that the same holds for x too by adopting the fibration method similarly to harpaz wittenberg

families of rationally connected varieties recall that a proper variety x is said to be rationally connected if two general points p q x are contained in the image of a map g p x this is clearly a birationally invariant property when x is smooth this turns out to be equivalent to the a priori weaker condition that two general points can be joined by a chain of rational curves and also to the a priori stronger condition that for any finite subset x there is a map g p x whose image contains and such that gtx is an ample bundle

on the fibration method for zero cycles and rational points conjectures on the existence of zero cycles on arbitrary smooth projective varieties over number elds were proposed by colliot th el ene sansuc kato and saito in the 1980 s we prove that these conjectures are compatible with brations for brations into rationally connected varieties over a curve in particular they hold for the total space of families of homogeneous spaces of linear groups with connected geometric stabilisers we prove the analogous result for rational points conditionally on a conjecture on locally split values of polynomials which a recent work of matthiesen establishes in the case of linear polynomials over the rationals

transmit beamforming in multiple antenna systems with finite rate feedback a vq based approach this paper investigates quantization methods for feeding back the channel information through a low rate feedback channel in the context of multiple input single output miso systems we propose a new quantizer design criterion for capacity maximization and develop the corresponding iterative vector quantization vq design algorithm the criterion is based on maximizing the mean squared weighted inner product mswip between the optimum and the quantized beamforming vector the performance of systems with quantized beamforming is analyzed for the independent fading case this requires finding the density of the squared inner product between the optimum and the quantized beamforming vector which is obtained by considering a simple approximation of the quantization cell the approximate density function is used to lower bound the capacity loss due to quantization the outage probability and the bit error probability the resulting expressions provide insight into the dependence of the performance of transmit beamforming miso systems on the number of transmit antennas and feedback rate computer simulations support the analytical results and indicate that the lower bounds are quite tight

faces and traces of the unit ball of a symmetric gauge function abstract we investigate the faces of the closed unit ball bg x g x 1 of a symmetric gauge function g r n r firstly we describe the group of symmetries of a face f of bg in terms of its barycenter we introduce the concept of standard face of bg i e a face whose relative interior meets the set d x r n x 1 x n 0 called here the standard cone of r n and the concept of trace of a face f which is the intersection of f with d characterizations of standard faces are given and we show that the trace of a face if nonempty is the trace of a standard face it is proved that a face f and its trace have the same dimension iff f is a standard face a relation is found between the dimension of f and the dimension of its symmetry core i e the set of points of f that are fixed by all symmetries of f other results of this type are given the main result has a long and delicate proof it completely characterizes in several different ways the subsets of bg that are traces of faces of bg

translation invariant maps and applications in this paper we study translation invariant maps on a linear space a method for estimating quantities induced by such maps is presented applications for differences of operators commutators g contractive maps and gruss type inequalities are also given

on g invariant norms abstract a result of r mathias and horn cf linear algebra appl 142 1990 63 on the representation of the unitarily invariant norm is extended in the context of eaton triples and of real semisimple lie algebras the representation is related to a function criteria for being a norm is given in terms of and the longest element of the underlying finite reflection group in particular for the real simple lie algebras is a norm on its cartan subspace for a given nonzero if and only if 0 where 0 is the longest element of the weyl group this is the case if the weyl group contains id some related results are obtained

grassmannian beamforming for multiple input multiple output wireless systems transmit beamforming and receive combining are simple methods for exploiting the significant diversity that is available in multiple input multiple output mimo wireless systems unfortunately optimal performance requires either complete channel knowledge or knowledge of the optimal beamforming vector both are hard to realize in this article a quantized maximum signal to noise ratio snr beamforming technique is proposed where the receiver only sends the label of the best beamforming vector in a predetermined codebook to the transmitter by using the distribution of the optimal beamforming vector in independent and identically distributed rayleigh fading matrix channels the codebook design problem is solved and related to the problem of grassmannian line packing the proposed design criterion is flexible enough to allow for side constraints on the codebook vectors bounds on the codebook size are derived to guarantee full diversity order results on the density of grassmannian line packings are derived and used to develop bounds on the codebook size given a capacity or snr loss monte carlo simulations are presented that compare the probability of error for different quantization strategies

norm hull of vectors and matrices let v be a real or complex finite dimensional vector space and let n be a set of norms on v the norm hull of a vector x v with respect to n is the set of vectors y v that satisfy y x for all n we study and give characterization of the norm hull for some sets of well known norms on general vector spaces and for the set of algebra norms and the set of induced norms on the algebra of n n real or complex matrices

on the norm property of g c radii and eaton triples abstract let v be a euclidean space and o v be the orthogonal group the norm criterion of the g c radius is studied in the framework of an eaton triple v g d

on beamforming with finite rate feedback in multiple antenna systems we study a multiple antenna system where the transmitter is equipped with quantized information about instantaneous channel realizations assuming that the transmitter uses the quantized information for beamforming we derive a universal lower bound on the outage probability for any finite set of beamformers the universal lower bound provides a concise characterization of the gain with each additional bit of feedback information regarding the channel using the bound it is shown that finite information systems approach the perfect information case as t 1 2 sup b t 1 where b is the number of feedback bits and t is the number of transmit antennas the geometrical bounding technique used in the proof of the lower bound also leads to a design criterion for good beamformers whose outage performance approaches the lower bound the design criterion minimizes the maximum inner product between any two beamforming vectors in the beamformer codebook and is equivalent to the problem of designing unitary space time codes under certain conditions finally we show that good beamformers are good packings of two dimensional subspaces in a 2t dimensional real grassmannian manifold with chordal distance as the metric