map of discrete system into continuous continuous limits of discrete systems with long range interactions are considered the map of discrete models into continuous medium models is defined a wide class of long range interactions that give the fractional equations in the continuous limit is discussed the one dimensional systems of coupled oscillators for this type of long range interactions are considered the discrete equations of motion are mapped into the continuum equation with the riesz fractional derivative

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

anisotropic fractal media by vector calculus in non integer dimensional space a review of different approaches to describe anisotropic fractal media is proposed in this paper differentiation and integration non integer dimensional and multi fractional spaces are considered as tools to describe anisotropic fractal materials and media we suggest a generalization of vector calculus for non integer dimensional space by using a product measure method the product of fractional and non integer dimensional spaces allows us to take into account the anisotropy of the fractal media in the framework of continuum models the integration over non integer dimensional spaces is considered in this paper differential operators of first and second orders for fractional space and non integer dimensional space are suggested the differential operators are defined as inverse operations to integration in spaces with non integer dimensions non integer dimensional space that is product of spaces with different dimensions allows us to give continuum models for anisotropic type of the media the poisson s equation for fractal medium the euler bernoulli fractal beam and the timoshenko beam equations for fractal material are considered as examples of application of suggested generalization of vector calculus for anisotropic fractal materials and media

a method for analysis of c 1 continuity of subdivision surfaces a sufficient condition for c1 continuity of subdivision surfaces was proposed by reif comput aided geom design 12 1995 pp 153 174 and extended to a more general setting in d zorin constr approx accepted for publication in both cases the analysis of c1 continuity is reduced to establishing injectivity and regularity of a characteristic map in all known proofs of c1 continuity explicit representation of the limit surface on an annular region was used to establish regularity and a variety of relatively complex techniques were used to establish injectivity we propose a new approach to this problem we show that for a general class of subdivision schemes regularity can be inferred from the properties of a sufficiently close linear approximation and injectivity can be verified by computing the index of a curve an additional advantage of our approach is that it allows us to prove c1 continuity for all valences of vertices rather than for an arbitrarily large but finite number of valences as an application we use our method to analyze c1 continuity of most stationary subdivision schemes known to us including interpolating butterfly and modified butterfly schemes as well as the kobbelt s interpolating scheme for quadrilateral meshes

isospectral graphs with identical nodal counts according to a recent conjecture isospectral objects have different nodal count sequences gnutzmann et al 2005 j phys a math gen 38 8921 33 we study generalized laplacians on discrete graphs and use them to construct the first non trivial counterexamples to this conjecture in addition these examples demonstrate a surprising connection between isospectral discrete and quantum graphs

continuous limit of discrete systems with long range interaction discrete systems with long range interactions are considered continuous medium models as continuous limit of discrete chain system are defined long range interactions of chain elements that give the fractional equations for the medium model are discussed the chain equations of motion with long range interaction are mapped into the continuum equation with the riesz fractional derivative we formulate the consistent definition of continuous limit for the systems with long range interactions in this paper we consider a wide class of long range interactions that give fractional medium equations in the continuous limit the power law interaction is a special case of this class

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

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

long range cohesive interactions of non local continuum faced by fractional calculus abstract a non local continuum model including long range forces between non adjacent volume elements has been studied in this paper the proposed continuum model has been obtained as limit case of two fully equivalent mechanical models i a volume element model including contact forces between adjacent volumes as well as long range interactions distance decaying between non adjacent elements ii a discrete point spring model with local springs between adjacent points and non local springs with distance decaying stiffness connecting non adjacent points under the assumption of fractional distance decaying interactions between non adjacent elements a fractional differential equation involving marchaud type fractional derivatives has been obtained for unbounded domains it is shown that for unbounded domains the two mechanical models revert to lazopoulos and eringen model with fractional distance decaying functions it has also been shown that for a confined bar the stress strain relation is substantially different from that obtained simply using the truncated marchaud derivatives since a double integral instead of convolution integral appears moreover in the analysis of bounded domains the governing equations turn out to an integro differential equation including only the integral part of marchaud fractional derivatives on finite interval the mechanical boundary condition for the proposed model has been introduced consistently on the basis of mechanical considerations and the constitutive law of the proposed continuum model has been reported by mathematical induction several numerical applications have been reported to show verify and assess the concepts listed in this paper

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

injectivity of real polynomial maps and łojasiewicz exponents at infinity in this paper we give a lower bound for the ojasiewicz exponent at infinity of a special class of polynomial maps mathbb r n to mathbb r s s 1 as a consequence we detect a class of polynomial maps mathbb r n to mathbb r n that are global diffeomorphisms if their jacobian determinant never vanishes

on globally diffeomorphic polynomial maps via newton polytopes and circuit numbers in this article we analyze the global diffeomorphism property of polynomial maps f mathbb r n rightarrow mathbb r n by studying the properties of the newton polytopes at infinity corresponding to the sum of squares polynomials vert f vert 2 2 this allows us to identify a class of polynomial maps f for which their global diffeomorphism property on mathbb r n is equivalent to their jacobian determinant det jf vanishing nowhere on mathbb r n in other words we identify a class of polynomial maps for which the real jacobian conjecture which was proven to be false in general still holds

lattice with long range interaction of power law type for fractional non local elasticity abstract lattice models with long range interactions of power law type are suggested as a new type of microscopic model for fractional non local elasticity using the transform operation we map the lattice equations into continuum equation with riesz derivatives of non integer orders the continuum equations that are obtained from the lattice model describe fractional generalization of non local elasticity models particular solutions and correspondent asymptotic of the fractional differential equations for displacement fields are suggested for the static case

well posedness for semidiscrete fractional cauchy problems with finite delay abstract we address the study of well posedness on lebesgue spaces of sequences for the following fractional semidiscrete model with finite delay 0 1 u n t u n u n f n n n 0 1 r n 0 where t is a bounded linear operator defined on a banach space x typically a space of functions like l p 1 p and corresponds to the time discretization of the continuous riemann liouville fractional derivative by means of the poisson distribution we characterize the existence and uniqueness of solutions in vector valued lebesgue spaces of sequences of the model 0 1 in terms of boundedness of the operator valued symbol z 1 z 1 i z t 1 z 1 z 1 whenever 0 1 and x satisfies a geometrical condition for this purpose we use methods from operator valued fourier multipliers and resolvent operator families associated to the homogeneous problem we apply this result to show a practical and computational criterion in the context of hilbert spaces

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 isospectral fruits of representation theory quantum graphs and drums we present a method which enables one to construct isospectral objects such as quantum graphs and drums one aspect of the method is based on representation theory arguments which are shown and proved the complementary part concerns techniques of assembly which are both stated generally and demonstrated for this purpose quantum graphs are grist to the mill we develop the intuition that stands behind the construction as well as the practical skills of producing isospectral objects we discuss the theoretical implications which include sunada s theorem of isospectrality sunada 1985 ann math 121 169 arising as a particular case of this method a gallery of new isospectral examples is presented and some known examples are shown to result from our theory

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

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

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

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

equivalence between isospectrality and isolength spectrality for a certain class of planar billiard domains isospectrality of the planar domains which are obtained by successive unfolding of a fundamental building block is studied in relation to isolength spectrality of the corresponding domains although an explicit and exact trace formula such as poisson s summation formula or selberg s trace formula is not known to exist for such planar domains equivalence between isospectrality and isolength spectrality in a certain setting can be proved by employing the matrix representation of transplantation of eigenfunctions as an application of the equivalence transplantable pairs of domains which are all isospectral pair of planar domains and therefore counter examples of kac s question can one hear the shape of a drum are numerically enumerated and it is found that at least up to the domain composed of 13 building blocks transplantable pairs coincide with those constructed by the method due to sunada

on the uniform observability of discrete time nonlinear systems in constructing an observer for a discrete time nonlinear system the system is commonly required to satisfy a certain kind of uniform observability condition that is the state should always be reconstructible from observation windows of a specific length irrespective of the values of the state and inputs in this technical note it is proved that this uniform requirement is unnecessary in the sense that if the initial state and inputs are on a compact set then the uniform observability is derived from its non uniform counterpart

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

finite geometries and diffractive orbits in isospectral billiards several examples of pairs of isospectral planar domains have been produced in the two dimensional euclidean space by various methods we show that all these examples rely on the symmetry between points and blocks in finite projective spaces from the properties of these spaces one can derive a relation between green s functions as well as a relation between diffractive orbits in isospectral billiards

lattice fractional calculus integration and differentiation of non integer orders for n dimensional physical lattices with long range particle interactions are suggested the proposed lattice fractional derivatives and integrals are represented by kernels of lattice long range interactions such that their fourier series transformations have a power law form with respect to components of wave vector continuous limits for these lattice fractional derivatives and integrals give the continuum derivatives and integrals of non integer orders with respect to coordinates lattice analogs of fractional differential equations that include suggested lattice differential and integral operators can serve as an important element of microscopic approach to nonlocal continuum models in mechanics and physics

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

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

can one hear the shape of a drum revisited a famous inverse problem posed by m kac can one hear the shape of a drum is concerned with isospectrality of drums or planer billiards and the first counter example was constructed by gordon webb and wolpert 1992 invent math 110 1 here we present pieces of numerical evidence showing that one can distinguish isospectral drums by measuring the scattering poles of exterior neumann problems this is based on the observation that the fredholm determinant appearing in the boundary element method admits a factorization into interior and exterior parts

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

a generalized model of elastic foundation based on long range interactions integral and fractional model abstract the common models of elastic foundations are provided by supposing that they are composed by elastic columns with some interactions between them such as contact forces that yield a differential equation involving gradients of the displacement field in this paper a new model of elastic foundation is proposed introducing into the constitutive equation of the foundation body forces depending on the relative vertical displacements and on a distance decaying function ruling the amount of interactions different choices of the distance decaying function correspond to different kind of interactions and foundation behavior the use of an exponential distance decaying function yields an integro differential model while a fractional power law decay of the distance decaying function yields a fractional model of elastic foundation ruled by a fractional differential equation it is shown that in the case of exponential decaying function the integral equation represents a model in which all the gradients of the displacement function appear while the fractional model is an intermediate model between integral and gradient approaches a fully equivalent discrete point spring model of long range interactions that may be used for the numerical solution of both integral and fractional differential equation is also introduced some green s functions of the proposed model have been included in the paper and several numerical results have been also reported to highlight the effects of long range forces and the governing parameters of the linear elastic foundation proposed

isospectral drums in involution graphs and euclidean ti domains the widely investigated question can one hear the shape of a drum which kac posed in his published lecture kac 1966 am math mon 73 1 23 was eventually answered negatively in gordon et al 1992 invent math 110 1 22 by construction of isospectral pairs in up to present all known planar counter examples are constructed by a certain tiling method and in this communication we call such examples isospectral euclidean ti domains from counter examples of this type one can construct a pair of finite involution graphs in this communication we address the question as to how the isospectrality of the domains for the laplacian influences the cospectrality of the involution graphs

on the boundary element method for billiards with corners the boundary element method is one of the reliable numerical schemes to solve the eigenvalue problem of the helmholtz equation which is justified by the fredholm theory for domains with a smooth boundary when a domain has corners however the corresponding integral equation is singular so that the boundary element method lacks its well established base employing a cutoff technique we here formulate a well grounded version of the boundary element method and also give a certain justification to the standard boundary element method even for domains with corners

on polynomial submersions of degree 4 and the real jacobian conjecture in r 2 abstract we prove the following version of the real jacobian conjecture let f p q r 2 r 2 be a polynomial map with nowhere zero jacobian determinant if the degree of p is less than or equal to 4 then f is injective the approach to prove this result leads to a complete classification up to affine change of coordinates of the polynomial submersions of degree 4 in r 2 whose level sets are not all connected

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

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

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

the cauchy problem for discrete time fractional evolution equations abstract this paper is devoted to the study of discrete time fractional evolution equations involving the riemann liouville like difference operator based on the relationship between c 0 semigroups and a distinguished class of sequences of operators we show the structure of the solutions for the inhomogeneous cauchy problem of abstract fractional difference equations further we establish two criteria for the existence and uniqueness of solutions for the semilinear cauchy problem some examples are also provided to illustrate our main results

resolving isospectral drums by counting nodal domains several types of systems have been put forward during the past few decades to show that there exist isospectral systems which are metrically different one important class consists of laplace beltrami operators for pairs of flat tori in r n with n 4 we propose that the spectral ambiguity can be resolved by comparing the nodal sequences the numbers of nodal domains of eigenfunctions arranged by increasing eigenvalues in the case of isospectral flat tori in four dimensions where a four parameter family of isospectral pairs is known we provide heuristic arguments supported by numerical simulations to support the conjecture that the isospectrality is resolved by the nodal count thus one can count the shape of a drum if it is designed as a flat torus in four dimensions less

nodal domains on isospectral quantum graphs the resolution of isospectrality we present and discuss isospectral quantum graphs which are not isometric these graphs are the analogues of the isospectral domains in which were introduced recently in gordon et al 1992 bull am math soc 27 134 8 chapman 1995 am math mon 102 124 buser et al 1994 int math res not 9 391 400 okada and shudo 2001 j phys a math gen 34 5911 22 jakobson et al 2006 j comput appl math 194 141 55 and levitin et al 2006 j phys a math gen 39 2073 82 all based on sunada s construction of isospectral domains sunada t 1985 ann math 121 196 86 after presenting some of the properties of these graphs we discuss a few examples which support the conjecture that by counting the nodal domains of the corresponding eigenfunctions one can resolve the isospectral ambiguity

chaotic vibration bifurcation stabilization and synchronization control for fractional discrete time systems abstract the main intention of this paper is to investigate the dynamics of fractional cubic map and fractional holmes map from four perspectives chaos bifurcation stabilization and synchronization via jacobian matrix algorithm we estimate the largest lyapunov exponents of fractional maps to examine the existence of chaos in accordance with the observation of chaotic vibration affected by fractional difference order in addition with the help of the novel asymptotic stability criterion for linear incommensurate fractional difference systems the effective control laws of stabilization and synchronization for fractional chaotic maps are achieved respectively besides experimental investigations and numerical simulations are implemented to further verify the effectiveness of the proposed results

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

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

global asymptotic stability for differentiable vector fields of r2 a let x r2 r2 be a differentiable map not necessarily c1 and let spec x be the set of complex eigenvalues of the derivative dxp when p varies in r2 if for some e 0 spec x 0 e then x is injective b let x r2 r2 be a differentiable vector field such that x 0 0 and spec x z c r z 0 then for all p r2 there is a unique positive trajectory starting at p moreover the limit set of p is equal to 0

maps that are roots of power series we introduce a class of polynomial maps that we call polynomial roots of powerseries and show that automorphisms with this property generate the automorphism group in any dimension in particular we determine generically which polynomial maps that preserve the origin are roots of powerseries we study the one dimensional case in greater depth

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

toward lattice fractional vector calculus an analog of fractional vector calculus for physical lattice models is suggested we use an approach based on the models of three dimensional lattices with long range inter particle interactions the lattice analogs of fractional partial derivatives are represented by kernels of lattice long range interactions where the fourier series transformations of these kernels have a power law form with respect to wave vector components in the continuum limit these lattice partial derivatives give derivatives of non integer order with respect to coordinates in the three dimensional description of the non local continuum the fractional differential operators have the form of fractional partial derivatives of the riesz type as examples of the applications of the suggested lattice fractional vector calculus we give lattice models with long range interactions for the fractional maxwell equations of non local continuous media and for the fractional generalization of the mindlin and aifantis continuum models of gradient elasticity

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

a jacobian criterion for the simultaneous injectivity on positive variables of linearly parameterized polynomial maps abstract consider a map g r r r n r p r m such that for k r r and x r n g k x l x f k x where l r n r p is a linear map and f k x i 1 r k i x a i v i i 1 r k i x 1 a i 1 x n a in v i a i z 0 n and v i r m are fixed for i 1 r we prove that the partially evaluated map g k r n r p r m is injective on r 0 n for every k r 0 r if and only if for each k r 0 r and x r 0 n the linear derivative map d g k x r n r p r m of g k at x is injective this result is useful for studying the uniqueness or multiplicity of equilibria in conservative systems of chemical reactions under mass action a map such as f would represent the rates of change of concentrations of all or some judiciously selected species the linear map l would represent the time invariant total concentrations to illustrate this application we prove the uniqueness of equilibria in a common pharmacological model of receptor ligand interaction without a customary assumption on rate constants that lets all equilibria be of a strong type known as detailed balance our result extends a theorem of craciun and feinberg applicable to maps of the kind of f that earlier result is directly applicable to models of chemical reactions that include the outflow of all species

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

diffractive orbits in isospectral billiards isospectral domains are non isometric regions of space for which the spectra of the laplace beltrami operator coincide in the two dimensional euclidean space instances of such domains have been given it has been proved for these examples that the length spectrum that is the set of the lengths of all periodic trajectories coincides as well however there is no one to one correspondence between the diffractive trajectories it will be shown here how the diffractive contributions to the green functions match nevertheless in a one to three correspondence

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

fractional liouville equation on lattice phase space abstract in this paper we propose a lattice analog of phase space fractional liouville equation the liouville equation for phase space lattice with long range jumps of power law types is suggested we prove that the continuum limit transforms this lattice equation into liouville equation with conjugate riesz fractional derivatives of non integer orders with respect to coordinates of continuum phase space an application of the fractional liouville equation with these riesz fractional derivatives to describe properties of plasma like nonlocal media is considered

fractional order difference equations for physical lattices and some applications fractional order operators for physical lattice models based on the grunwald letnikov fractional differences are suggested we use an approach based on the models of lattices with long range particle interactions the fractional order operators of differentiation and integration on physical lattices are represented by kernels of lattice long range interactions in continuum limit these discrete operators of non integer orders give the fractional order derivatives and integrals with respect to coordinates of the grunwald letnikov types as examples of the fractional order difference equations for physical lattices we give difference analogs of the fractional nonlocal navier stokes equations and the fractional nonlocal maxwell equations for lattices with long range interactions continuum limits of these fractional order difference equations are also suggested

the mechanically based approach to 3d non local linear elasticity theory long range central interactions abstract this paper presents the generalization to a three dimensional 3d case of a mechanically based approach to non local elasticity theory recently proposed by the authors in a one dimensional 1d case the proposed model assumes that the equilibrium of a volume element is attained by contact forces between adjacent elements and by long range forces exerted by non adjacent elements specifically the long range forces are modelled as central body forces depending on the relative displacement between the centroids of the volume elements measured along the line connecting the centroids further the long range forces are assumed to be proportional to a proper material dependent distance decaying function and to the products of the interacting volumes consistently with the modelling of the long range forces as central body forces the static boundary conditions enforced on the free surface of the solid involve only local stress due to contact forces the proposed 3d formulation is developed both in a mechanical and in a variational context for this the elastic energy functionals of the solid with long range interactions are introduced based on the principle of virtual work to set the proper correspondence between the mechanical and the kinematic variables of the model numerical applications are reported for 2d solids under plane stress conditions

the 5 canonical system on 3 folds of general type let x be a projective minimal gorenstein 3 fold of general type with canonical singularities we prove that the 5canonical map is birational onto its image

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