on global diffeomorphisms of euclidian space as is well known the first attempt at characterizing global diffeomorphisms of ir goes back to the 1906 work of hadamard l since that time progress in the global diffeomorphism problem has been rather slow notable contributions are those of levy 2 who generalized hadamard s theorem to banach spaces and of banach and mazur 3 and caccioppoli 4 who proved the so called properness criterion saying that a proper local homeomorphism of a banach space is a global homeomorphism the standard proofs of the theorems of hadamard and levy and of banach mazur and caccioppoli rely upon ideas of what is now called covering space theory via the important homotopy lifting property of coverings see e g 5 this approach makes it clear that it is only lack of compactness of ir that allows for the existence of noninjective local homeomorphisms when n l 2 the complex exponential when n 2 being the most famous example the theorems mentioned above were found again by various authors sometimes in special cases and sometimes in a different framework but always through some variant of the same argument a synthesis was made by rheinboldt 6 in 1969 part of his results were rediscovered five years later by plastock 7 and given a somewhat simpler exposition for the special case of banach spaces other contributions were made by meisters and olech 8 global homeomorphisms of bounded domains in i meyer 9 complementing the work of levy and radulescu and radulescu lo expanding upon plastock s results many other articles have been devoted to special cases where extra hypotheses are introduced consistently with the problem at hand for instance polynomial mappings only are relevant to the jacobian conjecture of algebraic geometry and hypotheses about the spectrum of the derivative at each point come up in a natural way in global stability questions for differential equations the new results presented in this paper have been obtained through arguments from critical point theory and differential topology here the central notion is that of admissible flow given a mapping f ii ir the main property of what we call an f admissible flow q ir x ii f o i f o is that for every x e ir f o the function t jf q t x l where 1 1 is the euclidean norm is strictly increasing and tends to 0 resp 00 as t tends to co resp 00 the basic feature we prove is that a local diffeomorphism f is a global one if and only if f o zi and there is an f admissible flow this statement makes it clear that 0 plays a special role regarding f admissibility and the condition f o 0 cannot be eliminated as examples will show the introduction of f admissibility followed the author s remark that in hilbert space hadamard s theorem could be derived from the mountain pass theorem of ambrosetti and

a generalization of n matrices abstract ky fan defines an n matrix to be a matrix of the form a ti b b 0 t b where b is the spectral radius of b and is the maximum of the spectral radii of all principal submatrices of b of order n 1 in this paper we define the closure n 0 matrices of n matrices by letting t it is shown that if a z and a 1 a n certain inequalities of n matrices are shown to hold for n 0 matrices and a method for constructing an n matrix from an m matrix is given

bounds on the capacity of ofdm underspread frequency selective fading channels the analysis of the channel capacity in the absence of prior channel knowledge noncoherent channel has gained increasing interest in recent years yet the noncoherent channel capacity is still unknown for the general case in this paper we derive bounds on the capacity of the noncoherent underspread complex gaussian orthogonal frequency division multiplexing wide sense stationary channel with uncorrelated scattering under a peak power constraint or a constraint on the second and fourth moments of the transmitted signal these bounds are characterized only by the system signal to noise ratio snr and by a newly defined quantity termed effective coherence time intuitively the effective coherence time can be interpreted as the length of a block in the block fading model in which a system with the same snr will achieve the same capacity as in the analyzed channel unlike commonly used coherence time definitions it is shown that the effective coherence time depends on the snr and is a nonincreasing function of it we show that for low snr the capacity is proportional to the effective coherence time while for higher snr the coherent channel capacity can be achieved provided that the effective coherence time is large enough

polynomial maps and a conjecture of samuelson abstract let f r n r n be a c 1 mapping it was conjectured by samuelson that if the upper left hand principal minors of the jacobian of f do not vanish on r n then f is injective however in 1965 gale and nikaido gave a simple counterexample to the case n 2 in this paper we show that the samuelson conjecture is true for polynomial mappings from c n to c n furthermore we give a precise description of such maps

a hypersurace which has the abhyankar moh property let x be an algebraic subset of cn we say that x has the abhyankarmoh property a m p if for any polynomial embedding f x cn there exists a polynomial automorphism f of cn such that f is a restriction of this automorphism to the set x in 6 10 11 20 it was shown that if dim x is sufficiently small relatively to n and if x has nice singularities then x has the a m p and the problem of extending the polynomial embedding of x into cn has in general many solutions unfortunately if dim x is big enough in comparison with n the situation is more difficult in particular if x is a generic hypersurface of degree n in cn n 2 then the problem of extending the polynomial embedding of x into cn has no more than one solution see 8 abhyankar and moh proved see 1 2 that a line in c2 has the a m p it can be easily inferred from this that the cross k2 x c2 x1 x2 0 has the a m p the analogous questions for a hyperplane and the n cross kn x cn x1 xn 0 in the n space n 2 has been open cf 1 and conjecture 12 in 12 the question for a hyperplane is still open moreover in the case n 2 no example of a hypersurface which had the a m p has been known

bandwidth scaling for fading multipath channels we show that very large bandwidths on fading multipath channels cannot be effectively utilized by spread spectrum systems that in a particular sense spread the available power uniformly over both time and frequency the approach is to express the input process as an expansion in an orthonormal set of functions each localized in time and frequency the fourth moment of each coefficient in this expansion is then uniformly constrained we show that such a constraint forces the mutual information to 0 inversely with increasing bandwidth simply constraining the second moment of these coefficients does not achieve this effect the results suggest strongly that conventional direct sequence code division multiple access cdma systems do not scale well to extremely large bandwidths to illustrate how the interplay between channel estimation and symbol detection affects capacity we present results for a specific channel and cdma signaling scheme

characterizing injectivity of classes of maps via classes of matrices abstract we present a framework for characterizing injectivity of classes of maps on cosets of a linear subspace by injectivity of classes of matrices using our formalism we characterize injectivity of several classes of maps including generalized monomial and monotonic not necessarily continuous maps in fact monotonic maps are special cases of component wise affine maps further we study compositions of maps with a matrix and other composed maps in particular rational functions our framework covers classical injectivity criteria based on mean value theorems for vector valued maps and recent results obtained in the study of chemical reaction networks

an exponential formula for polynomial vector fields ii lie series exponential substituition and rooted trees abstract grobner s lie series and the exponential formula provide different explicit formulas for the flow generated by a finite dimensional polynomial vector field the present paper gives 1 a generalization of the lie series in case of non commuting variables called exponential substituition 2 a structural understanding of the three formulas and their mutual relationships in terms of rooted trees and 3 as a byproduct new results on the enumeration coding and statistics of different kinds of rooted trees

d log and formal flow for analytic isomorphisms of n space given a formal map f f1 fn of the form z higher order terms we give tree expansion formulas and as sociated algorithms for the d log of f and the formal flow ft the coefficients which appear in these formulas can be viewed as certain generalizations of the bernoulli numbers and the bernoulli polynomials moreover the coefficient polynomials in the formal flow formula coincide with the strict order polynomials in combina torics for the partially ordered sets induced by trees applications of these formulas to the jacobian conjecture are discussed

the jacobian conjecture linear triangularization for homogeneous polynomial maps in dimension three let k be a field of characteristic zero and f k3 k3 a polynomial map of the form f x h where h is homogeneous of degree d 2 we show that the jacobian conjecture is true for such mappings more precisely we show that if jh is nilpotent there exists an invertible linear map t such that t 1ht 0 h2 x1 h3 x1 x2 where the hi are homogeneous of degree d as a consequence of this result we show that all generalized druzkowski mappings f x h x1 l1d xn lnd where li are linear forms for all i and d 2 are linearly triangularizable if jh is nilpotent and rkjh 3

on the existence of global analytic conjugations for polynomial mappings of yagzhev type abstract consider a mappingf c n c nof the form identity plus a term with polynomial components that are homogeneous of the third degree and suppose that the jacobian determinant of this mapping is constant throughout c n polynomial mapping of yagzhev type as a stronger version of the classical jacobian conjecture the question has been posed whether for some values of c 0 there exists a global change of variables conjugation on c nthrough which the mapping fbecomes its linear part at the origin van den essen has recently produced a simple yagzhev mapping for which no suchpolynomialconjugation exists we show here that van den essen s example still admitsglobal analyticconjugations the question on the existence of global conjugations for general yagzhev maps is then still open

constellation design for the noncoherent mimo rayleigh fading channel at general snr constellation design for the noncoherent multiple input multiple output mimo block rayleigh fading channel is considered for general signal to noise ratios snrs starting from a given base unitary constellation of finite cardinality and using the cutoff rate expression as the design criterion input probabilities and per antenna amplitudes for the constellation points are obtained via a difference of convex programming formulation using the mutual information as a performance metric it is shown that the optimized constellations significantly outperform the base unitary designs from which they are obtained in the low medium snr regime and indeed they also similarly outperform the mutual information achieved by isotropically distributed unitary inputs for the continuous input channel i e the so called unitary space time capacity ustc at sufficiently high snrs the resulting mutual information coincides with that of the base unitary designs thus the optimum constellation design technique works over the entire range of snrs the bit energy spectral efficiency tradeoff of the optimized constellations are also obtained and these provide valuable insights on modulation and coding which are especially useful for wideband channels where the snr per degree of freedom is low

capacity results for block stationary gaussian fading channels with a peak power constraint a peak power limited single antenna block stationary gaussian fading channel is studied where neither the transmitter nor the receiver knows the channel state information but both know the channel statistics this model subsumes most previously studied gaussian fading models the asymptotic channel capacity in the high signal to noise ratio snr regime is first computed and it is shown that the behavior of the channel capacity depends critically on the channel model for the special case where the fading process is symbol by symbol stationary it is shown that the codeword length must scale at least logarithmically with snr in order to guarantee that the communication rate can grow logarithmically with snr with decoding error probability bounded away from one an expression for the capacity per unit energy is also derived furthermore it is shown that the capacity per unit energy is achievable using temporal on off signaling with optimally allocated on symbols where the optimal on symbol allocation scheme may depend on the peak power constraint

the jacobian conjecture reduction of degree and formal expansion of the inverse introduction i the jacobian conjecture 1 statement of the jacobian problem first observations 2 some history of the jacobian conjecture 3 faulty proofs 4 the use of stabilization and of formal methods ii the reduction theorem 1 notation 2 statement of the reduction theorem 3 reduction to degree 3 4 proof of the reduction theorem 5 r linearization and unipotent reduction 6 nilpotent rank 1 jacobians iii the formal inverse 1 notation 2 abhyankar s inversion formula 3 the terms gj 4 the tree expansion g lt a t lfptf 5 calculations references

the classification of some polynomial maps with nilpotent jacobians abstract in the paper we first classify all polynomial maps h of the following form h h 1 x 1 x 2 x n h 2 x 1 x 2 h 3 x 1 x 2 h n x 1 x 2 with jh nilpotent after that we generalize the structure of h to h h 1 x 1 x 2 x n h 2 x 1 x 2 h 3 x 1 x 2 h 1 h n x 1 x 2 h 1

classification of digital affine noncommutative geometries it is known that connected translation invariant n dimensional noncommutative differentials dxi on the algebra k x1 xn of polynomials in n variables over a field k are classified by commutative algebras v on the vector space spanned by the coordinates these data also apply to construct differentials on the heisenberg algebra spacetime with relations x x where is an antisymmetric matrix as well as to lie algebras with pre lie algebra structures we specialise the general theory to the field k f2 of two elements in which case translation invariant metrics i e with constant coefficients are equivalent to making v a frobenius algebra we classify all of these and their quantum levi civita bimodule connections for n 2 3 with partial results for n 4 for n 2 we find 3 inequivalent differential structures admitting 1 2 and 3 invariant metrics respectively for n 3 we find 6 differential structures admitting 0 1 2 3 4 7 invariant metrics respectively we give some examples for n 4 and general n surprisingly not all our geometries for n 2 have zero quantum riemann curvature quantum gravity is normally seen as a weighted sum over all possible metrics but our results are a step towards a deeper approach in which we must also sum over differential structures over f2 we construct some of our algebras and associated structures by digital gates opening up the possibility of digital geometry it is known that connected translation invariant n dimensional noncommutative differentials dxi on the algebra k x1 xn of polynomials in n variables over a field k are classified by commutative algebras v on the vector space spanned by the coordinates these data also apply to construct differentials on the heisenberg algebra spacetime with relations x x where is an antisymmetric matrix as well as to lie algebras with pre lie algebra structures we specialise the general theory to the field k f2 of two elements in which case translation invariant metrics i e with constant coefficients are equivalent to making v a frobenius algebra we classify all of these and their quantum levi civita bimodule connections for n 2 3 with partial results for n 4 for n 2 we find 3 inequivalent differential structures admitting 1 2 and 3 invariant metrics respectively for n 3 we find 6 differential structures admitting 0 1 2 3 4 7 invariant metrics respectively we give some

integer solution for linear complementarity problem we consider the problem of finding an integer solution to a linear complementarity problem we introduce the class i of matrices for which the corresponding linear complementarity problem has an integer complementary solution for every vector q for which it has a solution strong principal unimodularity forms a sufficient condition for inclusion in the class i it is also shown to be necessary for some well known classes of matrices including the class of positive semidefinite matrices this is used in deriving necessary and sufficient conditions for a convex quadratic program to have an integer optimal solution for all b and c for which it has an optimal solution characterizations are also derived for some other well known classes of matrices in linear complementarity to belong to the class i in the end a peeling algorithm for finding integer solution for linear complementarity problems is given and related results are derived

lipschitzian q matrices are p matrices in this note we show that lipschitzianq matrices arep matrices by obtaining a necessary condition on lipschitzianq0 matrices the sufficiency of this condition has also been established by the first two authors along with another coauthor murthy parthasarathy and sriparna 1995

quantum geometry of field extensions we introduce a new kind of topological gauge configuration or soliton associated to the extension c r of the real numbers to the complex ones these configurations describe zero curvature gauge fields and nontrivial cohomology over the real line but with a quantum choice of differential calculus in general the quantum differential 1 forms on the line with coordinate algebra k x are in correspondence with field extensions of k and the quantum cohomology detects the nontriviality of the extension

on the capacity and energy efficiency of training based transmissions over fading channels in this paper the capacity and energy efficiency of training based communication schemes employed for transmission over a priori unknown rayleigh block fading channels are studied initially the case in which the product of the estimate error and transmitted signal is assumed to be gaussian noise is considered in this case it is shown that bit energy requirements grow without bound as the signal to noise ratio snr goes to zero and the minimum bit energy is achieved at a nonzero snr value below which one should not operate the effect of the block length on both the minimum bit energy and the snr value at which the minimum is achieved is investigated flash training and transmission schemes are analyzed and shown to improve the energy efficiency in the low snr regime in the second part of this paper the capacity and energy efficiency of training based schemes are investigated when the channel input vector in each coherence block is subject to peak power constraints the capacity achieving input structure is characterized and the magnitude distribution of the optimal input is shown to be discrete with a finite number of mass points the capacity bit energy requirements and optimal resource allocation strategies are obtained through numerical analysis the improvements in energy efficiency when on off keying ook with fixed peak power and vanishing duty cycle is employed are studied

linearized polynomial maps over finite fields abstract we consider polynomial maps described by so called multivariate linearized polynomials these polynomials are defined using a fixed prime power say q linearized polynomials have no mixed terms considering invertible polynomial maps without mixed terms over a characteristic zero field we will only obtain up to a linear transformation of the variables triangular maps which are the most basic examples of polynomial automorphisms however over the finite field f q automorphisms defined by linearized polynomials have in general an entirely different structure namely we will show that the linearized polynomial maps over f q are in one to one correspondence with matrices having coefficients in a univariate polynomial ring over f q furthermore composition of polynomial maps translates to matrix multiplication implying that invertible linearized polynomial maps correspond to invertible matrices this alternate description of the linearized polynomial automorphism subgroup leads to the solution of many famous conjectures most notably the jacobian conjecture for this kind of polynomials and polynomial maps

almost n matrices and linear complementarity abstract an almost n matrix a is one with real entries whose determinant is positive and proper principal minors are negative in this paper we obtain some characterization results for almost n matrices that are similar to n matrices among other things we show that an almost n matrix of first category with n 4 belongs to the class of q matrices as an application we derive a new univalence result

the inverse function theorem and the jacobian conjecture for free analysis we establish an invertibility criterion for free polynomials and free functions evaluated on some tuples of matrices we show that if the derivative is nonsingular on some domain closed with respect to direct sums and similarity the function must be invertible thus as a corollary we establish the jacobian conjecture in this context furthermore our result holds for commutative polynomials evaluated on tuples of commuting matrices

channel coherence in the low snr regime channel capacity in the limit of vanishing signal to noise ratio snr per degree of freedom is known to be linear in snr for fading and nonfading channels regardless of channel state information at the receiver csir it has recently been shown that the significant engineering difference between the coherent and the noncoherent fading channels including the requirement of peaky signaling and the resulting spectral efficiency is determined by how the capacity limit is approached as snr tends to zero or in other words the sublinear term in the capacity expression in this paper we show that this sublinear term is determined by the channel coherence level which we define to quantify the relation between the snr and the channel coherence time this allows us to trace a continuum between the case with perfect csir and the case with no csir at all using this approach we also evaluate the performance of suboptimal training schemes

a question of rentschler and the dixmier problem the following theorem is the main result of the paper theorem let r and t be somewhat commutative algebras with the same holonomic number and let r t be an algebra homomorphism then every holonomic t module m is via a holonomic r module and has finite length as an r module when applied to the weyl algebra this result gives a positive answer to a question of rentschler in the important case where r d x and t d y are rings of differential operators on smooth irreducible algebraic affine varieties x and y of the same dimension the result means that holonomicity is preserved by twisting by an arbitrary algebra homomorphism a short proof is given of the well known fact that an affirmative solution to the dixmier problem whether every algebra endomorphism of the weyl algebra an is an automorphism implies the jacobian conjecture the dixmier problem has a positive answer if and only if the twisted from both sides a n bimodule an is simple for each to begin with it is not even clear whether it is finitely generated the theorem implies that it has finite length moreover the bimodule is holonomic thus simple subfactors of it have the least possible gelfand kirillov dimension

foliations and polynomial diffeomorphisms of mathbb r 3 let y f g h mathbb r 3 to mathbb r 3 be a c 2 map and let spec y denote the set of eigenvalues of the derivative dy p when p varies in mathbb r 3 we begin proving that if for some 0 spec y cap epsilon epsilon emptyset then the foliation mathcal f k with k in f g h made up by the level surfaces k constant consists just of planes as a consequence we prove a bijectivity result related to the three dimensional case of jelonek s jacobian conjecture for polynomial maps of mathbb r n

on an achievable rate of large rayleigh block fading mimo channels with no csi training based transmission over rayleigh block fading multiple input multiple output mimo channels is investigated as a training method a combination of a pilot assisted scheme and a biased signaling scheme is considered the achievable rates of successive decoding sd receivers based on the linear minimum mean squared error lmmse channel estimation are analyzed in the large system limit by using the replica method under the assumption of replica symmetry it is shown that negligible pilot information is best in terms of the achievable rates of the sd receivers in the large system limit the obtained analytical formulas of the achievable rates can improve the existing lower bound on the capacity of the mimo channel with no channel state information csi derived by hassibi and hochwald for all snrs the comparison between the obtained bound and a high snr approximation of the channel capacity derived by zheng and tse implies that the high snr approximation is unreliable unless quite high snr is considered energy efficiency in the low snr regime is also investigated in terms of the power per information bit required for reliable communication the required minimum power is shown to be achieved at a positive rate for the sd receiver with no csi whereas it is achieved in the zero rate limit for the case of perfect csi available at the receiver moreover numerical simulations imply that the presented large system analysis can provide a good approximation for not so large systems the results in this paper imply that sd schemes can provide a significant performance gain in the low to moderate snr regimes compared to conventional receivers based on one shot channel estimation

hessian nilpotent polynomials and the jacobian conjecture let z z 1 z n and let a n i 1 2 z 2 i be the laplace operator the main goal of the paper is to show that the well known jacobian conjecture without any additional conditions is equivalent to what we call the vanishing conjecture for any homogeneous polynomial p z of degree d 4 if m p m z 0 for all m 1 then m p m 1 z 0 when m 0 or equivalently m p m 1 z 0 when m 3 2 3 n 2 1 it is also shown in this paper that the condition m p m z 0 m 1 above is equivalent to the condition that p z is hessian nilpotent i e the hessian matrix hes p z 2 p z i z j is nilpotent the goal is achieved by using the recent breakthrough work of m de bondt a van den essen and various results obtained in this paper on hessian nilpotent polynomials some further results on hessian nilpotent polynomials and the vanishing conjecture above are also derived

images of commuting differential operators of order one with constant leading coefficients abstract we first study some properties of images of commuting differential operators of polynomial algebras of order one with constant leading coefficients we then propose what we call the image conjecture on these differential operators and show that the jacobian conjecture bass et al 1982 bcw van den essen 2000 e de bondt 2009 bo hence also the dixmier conjecture dixmier 1968 d and the vanishing conjecture zhao 2007 z3 of differential operators with constant coefficients are actually equivalent to certain special cases of the image conjecture a connection of the image conjecture and hence also the jacobian conjecture with multidimensional laplace transformations of polynomials is also discussed

quadratic tame generators problem of rank three abstract we give a complete classification of quadratic homogeneous polynomial maps h and keller maps of the form x h for which rk j h 3 over a field k of arbitrary characteristic in particular we show that such a keller map is tame when char k 2

the dixmier conjecture and the shape of possible counterexamples we establish a lower bound for the size of possible counterexamples of the dixmier conjecture we prove that b 15 where b is the minimum of the greatest common divisor of the total degrees of p and q where p q runs over the counterexamples of the dixmier conjecture 2013 elsevier inc

the jacobian conjecture and the extensions of polynomial embeddings let x be an algebraic subset of c we say that x has the abhyanka r moh property amp if for any polynomial embedding 4 x of there exists a polynomial au tomorphism b of c such that b is a restriction of this au tomorphism to the set x in 3 9 it was shown that if dim x is sufficiently small relatively to n and if x has nice singularities then x has the am p and the problem of extending the polynomial embedding of x into c has in general many solutions unfortunately if dim x is big enough in compar ison with n the situation is more difficult in particular if x is a generic hypersurface of degree n in n 2 then the problem of extending the polynomial embedding of x into c has no more than one solution see 5 abhyankar and mob proved see 1 that a line in c 2 has the amp it can be easily inferred from this that the c r o s s k 2 x g 2 xl x2 0 has the amp the analogous questions for a hyperplane and the n cross k x c x l x 0 in the n space n 3 are open cf 1 and conjecture 12 in 6 moreover in the case n 3 no example of a hypersurface which has the a m p is known on the other hand in 4 we showed that for every n 2 the smooth hypersurface x x c xl x 2 2 is any non zero complex constant does not have the amp we recall an important conjecture connected with this subject

asymptotic behavior of error exponents in the wideband regime in this paper we investigate the fundamental tradeoff between rate and bandwidth when a constraint is imposed on the error exponent specifically we consider both additive white gaussian noise awgn and rayleigh fading channels where the input symbols are assumed to have a peak constraint for the awgn channel model the optimal values of rz 0 and rz 0 are calculated where rz 1 b is the maximum rate at which information can be transmitted over a channel with bandwidth b when the error exponent is constrained to be greater than or equal to z the computation of rz 0 follows gallager s infinite bandwidth reliability function computation while the computation of rz 0 is new and parallels verdu s second order calculation for channel capacity based on these calculations we say that a sequence of input distributions is near optimal if both rz 0 and rz 0 are achieved we show that quaternary phase shift keying qpsk a widely used signaling scheme is near optimal within a large class of input distributions for the awgn channel similar results are also established for a fading channel where full channel side information csi is available at the receiver

note on the markus yamabe conjecture for gradient dynamical systems abstract let v r n r n be a c 1 vector field which has a singular point o and its linearization is asymptotically stable at every point of r n we say that the vector field v satisfies the markus yamabe conjecture if the critical point o is a global attractor of the dynamical system x v x in this note we prove that if v is a gradient vector field i e v f f c 2 then the basin of attraction of the critical point o is the whole r n thus implying the markus yamabe conjecture for this class of vector fields an analogous result for discrete dynamical systems of the form x m 1 f x m is proved

a sufficient condition in order that the real jacobian conjecture in r 2 holds let f f g r2 r2 be a polynomial map such that det df x y is different from zero for all x y r2 and f 0 0 0 0 we prove that for the injectivity of f it is sufficient to assume that the higher homogeneous terms of the polynomials ffx ggx and ffy ggy do not have real linear factors in common the proofs are based on qualitative theory of dynamical systems

decomposing samuelson maps abstract a samuelson map is here defined as a differentiable self mapping of n space with the property that the leading principal minors of its jacobian matrix vanish nowhere structure theorems are proved for samuelson maps that satisfy various simple conditions on the pivots arising from gaussian elimination of the jacobian matrix the theorems yield representations of a samuelson map as a unique composition of invertible maps that alter only a single coordinate

the linear dependence problem for power linear maps abstract let bl l 1 k be m nl complex matrices and let x l c n l l 1 k be complex vector variables we show that the components of the map h b 1 x 1 d 1 b k x k d k are linearly dependent over c if and only if det b 1 b 1 d 1 b k b k d k 0 where means the hadamard product x and x d denote the conjugate transpose and the dth hadamard power of a matrix x respectively connections are established between the homogenous dependence problem hdp n d which arises in the study of the jacobian conjecture and the dependence problem for power linear maps pldp n d an algorithm is given to compute counterexamples to pldp n d from those to hdp n d and counterexamples to pldp n 3 are obtained for all n 67

wideband fading channel capacity with training and partial feedback we consider the capacity of a wideband fading channel with partial feedback subject to an average power constraint the channel is modeled as a set of parallel independent block rayleigh fading subchannels with finite coherence time l channel uses the transmitter probes a subset of subchannels during each coherence time by transmitting pilot sequences for channel estimation for each subchannel probed one bit of feedback indicates whether or not the channel gain exceeds a threshold allowing transmission our problem is to optimize jointly the training both length and power number of subchannels probed probing bandwidth and feedback threshold to maximize the achievable rate lower bound on ergodic capacity taking into account the subchannel estimation error optimizing the probing bandwidth balances diversity against the quality of the subchannel estimate we show that the achievable rate increases as s log l where s is the signal to noise ratio and exceeds the capacity with impulsive signaling given by s when l exceeds a positive threshold value moreover the optimal probing bandwidth scales as sl log2 l in contrast without feedback the optimal probing bandwidth for the probing scheme scales as sl1 3 and the achievable rate converges to s where the gap diminishes as sl 1 3

matrices and the linear complementarity problem the paper is a collection of results on the linear complementarity problem q m the results are stated in terms of the matrix m

the nagata automorphism is wild it is proved that the well known nagata automorphism of the polynomial ring in three variables over a field of characteristic zero is wild that is it can not be decomposed into a product of elementary automorphisms

digital quantum groups we find and classify all bialgebras and hopf algebras or quantum groups of dimension 4 over the field f2 0 1 we summarize our results as a quiver where the vertices are the inequivalent algebras and there is an arrow for each inequivalent bialgebra or hopf algebra built from the algebra at the source of the arrow and the dual of the algebra at the target of the arrow there are 314 distinct bialgebras and among them 25 hopf algebras with at most one of these from one vertex to another we find a unique smallest noncommutative and noncocommutative one which is moreover self dual and resembles a digital version of uq sl2 we also find a unique self dual hopf algebra in one anyonic variable x4 0 for all our hopf algebras we determine the integral and associated fourier transform operator viewed as a representation of the quiver we also find all quasitriangular or universal r matrix structures on our hopf algebras these induce solutions of the yang baxter or braid relations in any representation

preclusion of switch behavior in networks with mass action kinetics we study networks taken with mass action kinetics and provide a jacobian criterion that applies to an arbitrary network to preclude the existence of multiple positive steady states within any stoichiometric class for any choice of rate constants we are concerned with the characterization of injective networks that is networks for which the species formation rate function is injective in the interior of the positive orthant within each stoichiometric class we show that a network is injective if and only if the determinant of the jacobian of a certain function does not vanish the function consists of components of the species formation rate function and a maximal set of independent conservation laws the determinant of the function is a polynomial in the species concentrations and the rate constants linear in the latter and its coefficients are fully determined the criterion also precludes the existence of degenerate steady states further we relate injectivity of a network to that of the network obtained by adding outflow or degradation reactions for all species

delay limited capacity in the low power regime outage performance of the m block fading with additive white gaussian noise bf awgn is investigated in the low power regime we consider delay constrained constant rate communications with perfect channel state information csi at both the transmitter and the receiver csi tr under a short term power constraint stpc and a long term power constraint ltpc subject to stpc we show that selection diversity that allocates all the power to the strongest block is asymptotically optimal then we provide a simple characterization of the outage probability in the regime of interest we quantify the reward due to csi tr over the constant rate constant power scheme and show that this reward increases with the delay constraint for instance for rayleigh fading we find that a power gain up to 4 3 db is achievable subject to ltpc we show that the above guidelines still holds and that the outage performance improves due to the flexibility of the ltpc over the stpc more interestingly we prove that ltpc allows zero outage communication even at low snr and characterize the delay limited capacity at low snr in a simple form more precisely we establish that the delay limited capacity scales linearly with the power constraint for a given m our framework highlights the benefit of fading at low snr as the delay limited capacity may outperform the awgn capacity for instance for rayleigh fading and with m 3 the delay limited capacity is 16 higher than the capacity of an awgn channel

on the length of polynomial automorphisms of the affine plane the automorphism group of the affine plane is a mysterious and challenging object although we know that it is an amalgamated product of two well known subgroups many questions are still unsolved moreover the group has the structure of an infinite dimensional algebraic group but the interactions between these two structures are not yet clear in this paper we study the length of an element defined using the amalgamated structure with respect to the algebraic structure if the ground field is of characteristic zero we prove that the length is a lower semicontinuous function on the group

ky fan s n matrices and linear complementarity problems we consider the linear complementarity problem lcp w az q w 0 z 0 wtz 0 when all the off diagonal entries ofa are nonpositive the class of z matrices all the proper principal minors ofa are positive and the determinant ofa is negative the class of almost p matrices we shall call this the class of f matrices we show that ifa is a z matrix thena is an f matrix if and only if lcp q a has exactly two solutions for anyq 0 q 0 and has at most two solutions for any otherq

dual operators and lagrange inversion in several variables the original lagrange inversion formula which gives explicitly the inverse under composition of a formal series was obtained by lagrange in 1770 through formal computations involving logarithms of inlmite products see lagrange 121 there exists an extensive literature on various versions of the inversion formula obtained by algebraic analytic and combinatorial methods comtet 3 gives a survey and an ample bibliography during the last ten years there have appeared several papers on generalizations and new proofs of the inversion formula in 1974 abhyankar obtained an inversion formula using algebraic methods see bass connell and wright 11 several authors have related lagrange inversion with rota s finite operator calculus 151 for example joni 8 garsia and joni 4 hofhauer 7 and roman and rota 14 joyal and labelle used a combinatorial theory of formal series to obtain inversion formulas see 9 l i and viskov 161 used ideas from lie algebra theory to prove formulas similar to abhyankar s in the present paper we use operator methods to obtain inversion formulas in several variables that generalize the formulas of abhyankar joni and viskov we get our results studying an algebra with involution of linear operators on the algebra of formal laurent series in several indeterminates we also show that by means of an anti isomorphism of operator algebras which we call the operator bore1 transform the study of finite operator calculus can be reduced to the study of certain groups of operators on the algebra of formal laurent series our approach may be considered as an algebraic analogue of the complex variable methods used by good 5 to obtain the lagrange inversion formula as a consequence of the change of variables theorem for the

on characterizing n matrices using linear complementarity we show that a square matrix a with at least one positive entry and all principal minors negative can be characterized in terms of the number of solutions the linear complementarity problem q a with the matrix a has for different vectors q such a matrix can also be characterized in terms of a sign nonreversal property these results complement the known results for a square matrix a 0 with all its principal minors negative

injectivity and almost global asymptotic stability of hurwitz vector fields abstract we present in dimension n 3 a survey of examples to the jacobian conjecture the weak markus yamabe conjecture furthermore we show and construct new examples of vector fields where the origin is almost globally asymptotic stable by using the novel concept of density functions introduced by rantzer

polynomial maps with nilpotent jacobians in dimension three abstract in this paper we first classify all polynomial maps of the form h u x y v x y z h x y v x y z in certain conditions then we classify some polynomial maps of the form h u x y v x y z h x y z

the strong factorial conjecture abstract in this paper we present an unexpected link between the factorial conjecture 8 and furter s rigidity conjecture 13 the factorial conjecture in dimension m asserts that if a polynomial f in m variables x i over c is such that l f k 0 for all k 1 then f 0 where l is the c linear map from c x 1 x m to c defined by l x 1 l 1 x m l m l 1 l m the rigidity conjecture asserts that a univariate polynomial map a x with complex coefficients of degree at most m 1 such that a x x mod x 2 is equal to x if m consecutive coefficients of the formal inverse for the composition of a x are zero

quadratic homogeneous keller maps of rank two abstract let h be a quadratic homogeneous polynomial map of dimension n over an infinite field in which 2 is invertible such that its jacobian jh is nilpotent meisters and olech have shown that jh is strongly nilpotent if n 4 they also proved that it is not true when n 5 we show that if rank j h 2 and n arbitrary then jh is strongly nilpotent we also give examples to show that this is no longer true for any rank and dimension as long as the rank is greater than 2

wideband fading channels with feedback the rayleigh flat fading channel at low snr is considered with full channel state information csi at the transmitter and receiver its capacity is shown to be essentially snr log 1snr nats symbol as snr goes to zero in fact this rate can be achieved with a just one bit of csi at the transmitter per fading realization and with no receiver csi the capacity for the case of noisy transmitter csi is also found then a rayleigh block fading channel of coherence interval t 1 snr is considered which has causal feedback and no a priori csi a training based scheme is proposed for such channels which achieves a rate of snr logt nats symbol in the limit of small snr and large t thus when coherence interval t is of the order 1 snr without any a priori csi at either end the capacity with full csi at both ends is achievable for smaller values of t a rate of snr logt nats symbol is shown to be achievable

on the algebraic identifiability of finite impulse response channels driven by linearly precoded signals it is common in wireless communications to perform some form of linear precodingoperation on the source signal prior to transmission over a channel although the traditional reason for precoding is to improve the performance of the communication system this paper draws attention to the fact that the receiver can identify the impulse response of the channel without any prior knowledge of the transmitted signal simply by solving a system of polynomial equations since different precoders lead to different systems of equations this paper addresses the fundamental issue of determiningwhich classes of linear precoders lead to a system of equations havinga unique solution in doingso basic properties of polynomial equations which are useful for studyingother identifiability issues commonly encountered in eng ineeringand the applied sciences are presented 2004 elsevier b v all rights reserved

topologies and structures of the cremona groups we study the algebraic structure of the n dimensional cremona group and show that it is not an algebraic group of innite dimension ind group if n 2 we describe the obstruction to this which is of a topological nature by contrast we show the existence of a euclidean topology on the cremona group which extends that of its classical subgroups and makes it a topological group

the capacity of discrete time memoryless rayleigh fading channels we consider transmission over a discrete time rayleigh fading channel in which successive symbols face independent fading and where neither the transmitter nor the receiver has channel state information subject to an average power constraint we study the capacity achieving distribution of this channel and prove it to be discrete with a finite number of mass points one of them located at the origin we numerically compute the capacity and the corresponding optimal distribution as a function of the signal to noise ratio snr the behavior of the channel at low snr is studied and finally a comparison is drawn with the ideal additive white gaussian noise channel

optimal constellations for the low snr noncoherent mimo block rayleigh fading channel reliable communication over the discrete input continuous output noncoherent multiple input multiple output mimo rayleigh block fading channel is considered when the signal to noise ratio snr per degree of freedom is low two key problems are posed and solved to obtain the optimum discrete input in both problems the average and peak power per space time slot of the input constellation are constrained in the first one the peak power to average power ratio ppapr of the input constellation is held fixed while in the second problem the peak power is fixed independently of the average power in the first ppapr constrained problem the mutual information which grows as o snr2 is maximized up to second order in snr in the second peak constrained problem where the mutual information behaves as o snr the structure of constellations that are optimal up to first order or equivalently that minimize energy per bit are explicitly characterized furthermore among constellations that are first order optimal those that maximize the mutual information up to second order or equivalently the wideband slope are characterized in both ppapr constrained and peak constrained problems the optimal constellations are obtained in closed form as solutions to nonconvex optimizations and interestingly they are found to be identical due to its special structure the common solution is referred to as space time orthogonal rank one modulation or storm in both problems it is seen that storm provides a sharp characterization of the behavior of noncoherent mimo capacity

a new class of invertible polynomial maps abstract in this paper we present a new large class of polynomial maps f x h a n a n definition 1 1 on every commutative ring a for which the jacobian conjecture is true in particular h does not need to be homogeneous we also show that for all h in this class satisfying h 0 0 the n th iterate h h 0

complete abelian integrals for polynomials whose generic fiber is biholomorphic to c abstract let h be a polynomial of degree m 1 on c 2 such that its generic fiber is biholomorphic to c and let be an arbitrary polynomial 1 form of degree n on c 2 we give an upper bound depending only on m and n for the number of isolated zeros of the complete abelian integral defined by h and

principal pivot transforms of some classes of matrices in this paper we study and characterize various classes of matrices that are defined based on principal pivot transforms we show that matrices in these classes have nonnegative principal minors

the lned and lfed conjectures for algebraic algebras abstract let k be a field of characteristic zero and a a k algebra such that all the k subalgebras generated by finitely many elements of a are finite dimensional over k a k e derivation of a is a k linear map of the form i for some k algebra endomorphism of a where i denotes the identity map of a in this paper we first show that for all locally finite k derivations d and locally finite k algebra automorphisms of a the images of d and i do not contain any nonzero idempotent of a we then use this result to show some cases of the lfed and lned conjectures proposed in 21 more precisely we show the lned conjecture for a and the lfed conjecture for all locally finite k derivations of a and all locally finite k e derivations of the form i with being surjective in particular both conjectures are proved for all finite dimensional k algebras furthermore some finite extensions of derivations and automorphism to inner derivations and inner automorphisms respectively have also been established this result is not only crucial in the proofs of the results above but also interesting on its own right

tame and wild coordinates of z x y abstract algorithms to decide whether a polynomial is a coordinates tame coordinate respectively of z x y are given moreover if a polynomial p z x y is a coordinate tame coordinate respectively the algorithms effectively construct a concrete automorphism of z x y that sends x to p the algorithm to determine tame coordinate is applicable to r x y where r is an euclidean domain while the algorithm to determine coordinate is applicable to d x y where d is a unique factorization domain

power linear keller maps with ditto triangularizations abstract we show that power linear keller maps f x 1 a 1 x d x 2 a 2 x d x n a n x d are linearly triangularizable if 1 rk a 2 or 2 cork a 2 and d 3 or 3 cork a 3 d 5 and the diagonal of a is nonzero furthermore we show that the triangularizations can be chosen power linear as well

an exponential formula for polynomial vector fields abstract the well known matrix exponentiation for linear first order systems of ode is generalised to polynomial vector fields the substitution process of our exponential formula is closely related to the usual iteration of polynomial mappings

communication on the grassmann manifold a geometric approach to the noncoherent multiple antenna channel we study the capacity of multiple antenna fading channels we focus on the scenario where the fading coefficients vary quickly thus an accurate estimation of the coefficients is generally not available to either the transmitter or the receiver we use a noncoherent block fading model proposed by marzetta and hochwald see ibid vol 45 p 139 57 1999 the model does not assume any channel side information at the receiver or at the transmitter but assumes that the coefficients remain constant for a coherence interval of length t symbol periods we compute the asymptotic capacity of this channel at high signal to noise ratio snr in terms of the coherence time t the number of transmit antennas m and the number of receive antennas n while the capacity gain of the coherent multiple antenna channel is min m n bits per second per hertz for every 3 db increase in snr the corresponding gain for the noncoherent channel turns out to be m 1 m t bits per second per hertz where m min m n t 2 the capacity expression has a geometric interpretation as sphere packing in the grassmann manifold

algorithmic detection and construction of n matrices abstract n matrices are real n n matrices all of whose principal minors are negative we provide i an o 2 n test to detect whether or not a given matrix is an n matrix and ii a characterization of n matrices leading to the recursive construction of every n matrix

incompatibility of diophantine equations arising from the strong factorial conjecture abstract in the present article we establish new evidence in support of the strong factorial conjecture of edo and van den essen 2 by proving it in several special cases for example we show that the conjecture holds for powers of linear forms and sums of prime powers of the variables finally we show one way of constructing new examples of polynomials satisfying the conjecture using known examples

on additive nilpotency of jacobian matrices of polynomial maps abstract let h c n c n be a polynomial map jh the jacobian matrix of h and v h the linear subspace of m n c spanned by j h c n the set of evaluated jacobian matrices of h we describe the dimension of v h through a power linear gorni zampieri mate of h and give a numerical algorithm for computing this dimension and determining whether jh is additive nilpotent i e whether v h is a nilpotent subspace