on very sparse circulant 0 1 matrices in this paper a particular partition on blocks of generalized circulant 0 1 matrices of composite order with three ones per row is determined using this result it is shown that the bipartite graph associated with a similar matrix in the usual way has genus 1 but a particular case where it holds 0 moreover a lower bound for the permanent of these matrices is achieved

rigidity for anosov actions of higher rank lattices this is the simplest example of a large class of analytic standard actions of lattices in semisimple lie groups on locally homogeneous spaces a basic problem is to understand the differentiable actions near to such a standard action in terms of their geometry and dynamics cf 13 50 51 a cr action so if x x x of a group f on a compact manifold x is said to be anosov if there exists at least one element yh e f such that p yh is an anosov diffeomorphism of x we begin in this paper to study the anosov differentiable actions of lattices including many standard algebraic examples and especially to study their stability properties our main theme is that either the cr rigidity or the cr deformation rigidity of an anosov action for 1 r cr or even for the real analytic case can be shown just by studying the behavior of the periodic orbits for the action there are two notions of structural stability that appear in this paper rigidity and deformation rigidity a cl perturbation of a cr action so is simply another cr action spl such that for a finite set of generators 81 ad of f the cr diffeomorphisms p 8i and pl 8i are cl close for all i an action so is said to be cr rigid or topologically rigid if r 0 if every sufficiently small c perturbation of so is cr conjugate to so for 0 r oo or r w in the case of real analytic actions a cl deformation of an action so is a continuous path of cr actions pt defined for some 0 t e with spo sd an action sd is said to be cr deform

topologies on measured groupoids abstract if an analytic borel group g has a quasiinvariant measure it is known that g is actually a locally compact group with the original borel structure being generated by the topology and the original measure being equivalent to haar measure in this paper a variation is given on the known proof which then extends to show that an analytic measured groupoid has a compact and also a locally compact inessential reduction which is a topological groupoid in the compact case it is proved that every almost homomorphism agrees a e with a strict homomorphism also the topology is used to show that every measured groupoid has a complete countable section 7 and that every locally compact equivalence relation has a complete transversal 3 these are further used to show that some results of feldman et al 7 apply in general and that a locally compact groupoid with continuous haar system has sufficiently many non singular borel g sets provided that the orbit measures are atom free 23

ergodic theory group representations and rigidity in these lectures we discuss some topics concerning the relationship of ergodic theory representation theory and the structure of lie groups and their discrete subgroups

on the number of different permanents of some sparse 0 1 circulant matrices abstract starting from known results about the number of possible values for the permanents of 0 1 circulant matrices with three nonzero entries per row and whose dimension n is prime we prove corresponding results for n power of a prime n product of two distinct primes and n 2 3 h supported by some experimental results we also conjecture that the number of different permanents of n n 0 1 circulant matrices with k nonzero per row is asymptotically equal to n k 2 k o n k 3

electrical characteristics of a class of n order triangular network abstract this paper investigates the electrical properties of a class of n order triangular resistor network model the authors establish by constructing equivalent model the second order nonlinear difference equation model we propose by using equivalent transformation method the general and specific methods of solution for the nonlinear difference equation and at once obtain an accurate equivalent resistance formula of n order triangular resistor network and a series of special conclusions are analyzed and compared finally we study the oscillation and resonance characteristics of the triangular lc network under different conditions where the method used is variable substitution by using the equivalent resistance results obtained this study reveals the electrical characteristics of triangular lc network and establishes a new theoretical foundation for related disciplines

on the permanent of certain 0 1 toeplitz matrices abstract we obtain convenient expressions and or efficient algorithms for the permanent of certain very sparse 0 1 toeplitz matrices the classes of matrices considered here include some nontrivial examples of circulants to which none of the previous approaches could be successfully applied

on prime factors of determinants of circulant matrices abstract we find the probability that the determinant of an integer circulant n n matrix is divisible by the prime p where p does not divide n

on the period spectrum of a symplectic mapping let m be a compact riemannian manifold it has been known for a long time that the singularities of the wave trace trace cos t are located at the periods of the closed geodesics do these singularities also contain information about the geometry of m in the neighborhood of a closed geodesic we prove that the birkhoff canonical form of the poincare map can be determined from the singularities of the wave trace

smooth conjugacy and s r b measures for uniformly and non uniformly hyperbolic systems we give a new proof of the fact that the eigenvalues at corresponding periodic orbits forms a complete set of invariants for the smooth conjugacy of low dimensional anosov systems we also show that if a homeomorphism conjugating two smooth low dimensional anosov systems is absolutely continuous then it is as smooth as the maps we furthermore prove generalizations of these facts for non uniformly hyperbolic systems as well as extensions and counterexamples in higher dimensions

on ulam von neumann transformations we define and study ulam von neumann transformations which are certain interval mappings and conjugate toq x 1 2x2 on 1 1 we use a singular metric on 1 1 to study a ulam von neumann transformation this singular metric is universal in the sense that it does not depend on any particular mapping but only on the exponent of this mapping at its unique critical point we give the smooth classification of ulam von neumann transformations by their eigenvalues at periodic points and exponents and asymmetries

finite size corrections and scaling for the triangular lattice dimer model with periodic boundary conditions we analyze the partition function of the dimer model on m x n triangular lattice wrapped on the torus obtained by fendley moessner and sondhi phys rev b 66 214513 2002 based on such an expression we then extend the algorithm of ivashkevich izmailian and hu j phys a 35 5543 2002 to derive the exact asymptotic expansion of the first and second derivatives of the logarithm of the partition function at the critical point and find that the aspect ratio dependence of finite size corrections and the finite size scaling functions are sensitive to the parity of the number of lattice sites along the lattice axis

recursion transform method for computing resistance of the complex resistor network with three arbitrary boundaries we develop a general recursion transform r t method for a two dimensional resistor network with a zero resistor boundary as applications of the r t method we consider a significant example to illuminate the usefulness for calculating resistance of a rectangular m ifmmode times else texttimes fi n resistor network with a null resistor and three arbitrary boundaries a problem never solved before since green s function techniques and laplacian matrix approaches are invalid in this case looking for the exact calculation of the resistance of a binary resistor network is important but difficult in the case of an arbitrary boundary since the boundary is like a wall or trap which affects the behavior of finite network in this paper we obtain several general formulas of resistance between any two nodes in a nonregular m ifmmode times else texttimes fi n resistor network in both finite and infinite cases in particular 12 special cases are given by reducing one of the general formulas to understand its applications and meanings and an integral identity is found when we compare the equivalent resistance of two different structures of the same problem in a resistor network

smooth classification of geometrically finite one dimensional maps the scaling function of a one dimensional markov map is defined and studied we prove that the scaling function of a non critical geometrically finite one dimensional map is holder continuous while the scaling function of a critical geometrically finite one dimensional map is discontinuous we prove that scaling functions determine lipschitz conjugacy classes and moreover that the scaling function and the exponents and asymmetries of a geometrically finite one dimensional map are complete c1 invariants within a mixing topological conjugacy class

dimer model on a triangular lattice we analyze the partition function of the dimer model on an mathcal m ifmmode times else texttimes fi mathcal n triangular lattice wrapped on a torus obtained by fendley moessner and sondhi phys rev b 66 214513 2002 from a finite size analysis we have found that the dimer model on such a lattice can be described by a conformal field theory having a central charge c ensuremath 2 the shift exponent for the specific heat is found to depend on the parity of the number of lattice sites mathcal n along a given lattice axis e g for odd mathcal n we obtain the shift exponent ensuremath lambda 1 while for even mathcal n it is infinite ensuremath lambda ensuremath infty in the former case therefore the finite size specific heat pseudocritical point is size dependent while in the latter case it coincides with the critical point of the thermodynamic limit

nonhomogeneous locally free actions of the affine group we classify locally free smooth actions of the real affine group on closed three dimensional manifolds up to smooth conjugacy in particular we show that there are non homogeneous actions when the manifold is not a rational homology sphere and its fundamental group is not solvable we say two right actions 1 and 2 of a lie group g on manifolds m1 and m2 are c conjugate if there exist an isomorphism of g and a c diffeomorphism h from m1 to m2 such that h 1 p g 2 h p g for any g 2 g and p 2 m1 the map h is called a c conjugacy between 1 and 2 we also say 1 and 2 are c orbit equivalent if there exists a c

two point resistance of a resistor network embedded on a globe we consider the problem of two point resistance in an m ensuremath 1 ifmmode times else texttimes fi n resistor network embedded on a globe a geometry topologically equivalent to an m ifmmode times else texttimes fi n cobweb with its boundary collapsed into one single point we deduce a concise formula for the resistance between any two nodes on the globe using a method of direct summation pioneered by one of us z z tan l zhou and j h yang j phys a math theor 46 195202 2013 this method is contrasted with the laplacian matrix approach formulated also by one of us f y wu j phys a math gen 37 6653 2004 which is difficult to apply to the geometry of a globe our analysis gives the result in the form of a single summation

the birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

theory of impedance networks the two point impedance and lc resonances we present a formulation of the determination of the impedance between any two nodes in an impedance network an impedance network is described by its laplacian matrix l which has generally complex matrix elements we show that by solving the equation lu u with orthonormal vectors ua the effective impedance between nodes p and q of the network is zpq u p u q 2 where the summation is over all not identically equal to zero and u p is the pth component of u for networks consisting of inductances l and capacitances c the formulation leads to the occurrence of resonances at frequencies associated with the vanishing of this curious result suggests the possibility of practical applications to resonant circuits our formulation is illustrated by explicit examples

an update on minc s survey of open problems involving permanents abstract we summarise the progress which has been made since 1986 on the conjectures and open problems listed in h minc s survey articles on the theory of permanents

the equivalent resistance of a 3 n cobweb network and its conjecture of an m n cobweb network we investigate the equivalent resistance of a 3 n cobweb network the difference equations of the model are constructed by network analysis and their general solution is obtained by matrix transformations it is found that the equivalent resistance can be expressed by the trigonometric function of k 7 which decreases with the increase of the order n by analyzing and comparing the equivalent resistances of the 1 n 2 n and 3 n cobweb networks a conjecture on the equivalent resistance of the m n cobweb network is proposed

exact evaluation of the resistance in an infinite face centered cubic network the equivalent resistance between the origin and the lattice site 2n 0 0 in an infinite face centered cubic fcc network consisting from identical resistors each of resistance r has been evaluated analytically and numerically the asymptotic behavior of the equivalent resistance has been also investigated finally some numerical values for the equivalent resistance are presented

perturbations of geodesic flows on surface of constant negative curvature and their mixing properties we consider one parameter analytic hamiltonian perturbations of the geodesic flows on surfaces of constant negative curvature we find two different necessary and sufficient conditions for the canonical equivalence of the perturbed flows and the non perturbed ones one condition says that the hamilton jacobi equation introduced in this work for the conjugation problem should admit a solution as a formal power series not necessarily convergent in the perturbation parameter the alternative condition is based on the identification of a complete set of invariants for the canonical conjugation problem the relation with the similar problems arising in the kam theory of the perturbations of quasi periodic hamiltonian motions is briefly discussed as a byproduct of our analysis we obtain some results on the livscic guillemin kazhdan equation and on the fourier series for the sl 2 group we also prove that the analytic functions on the phase space for the geodesic flow of unit speed have a mixing property with respect to the geodesic flow and to the invariant volume measure which is exponential with a universal exponent independent on the particular function equal to the curvature of the surface divided by 2 this result is contrasted with the slow mixing rates that the same functions show under the horocyclic flow in this case we find that the decay rate is the inverse of the time up to logarithms

theory of resistor networks the two point resistance the resistance between two arbitrary nodes in a resistor network is obtained in terms of the eigenvalues and eigenfunctions of the laplacian matrix associated with the network explicit formulae for two point resistances are deduced for regular lattices in one two and three dimensions under various boundary conditions including that of a mobius strip and a klein bottle the emphasis is on lattices of finite sizes we also deduce summation and product identities which can be used to analyse large size expansions in two and higher dimensions

computation of sparse circulant permanents via determinants we consider the problem of computing the permanent of circulant matrices we apply some recent results on the computation of the number of perfect matchings of small genus graphs in order to show that the permanent of a circulant matrix with three nonzero entries per row is the linear combination of just four determinants of circulant matrices with the same structure as the original matrix we also show that the same result holds true for a class of circulant matrices with four nonzero entries per row related to the dimer problem with periodic boundary conditions conversely we give hints at the fact that more general circulants do not share similar properties and thus should be dealt with by means of radically different approaches

lyapunov exponents and rigidity of anosov automorphisms and skew products abstract in this paper we obtain local rigidity results for linear anosov diffeomorphisms in terms of lyapunov exponents more specifically we show that given an irreducible linear hyperbolic automorphism l with simple real eigenvalues with distinct absolute values any small perturbation preserving the volume and with the same lyapunov exponents is smoothly conjugate to l we also obtain rigidity results for skew products over anosov diffeomorphisms given a volume preserving partially hyperbolic skew product diffeomorphism f 0 over an anosov automorphism of the 2 torus we show that for any volume preserving perturbation f of f 0 with the same average stable and unstable lyapunov exponents the center foliation is smooth

an asymptotic universal focal decomposition for non isochronous potentials galileo in the xvii century observed that the small os cillations of a pendulum seem to have constant period in fact the taylor expansion of the period map of the pendulum is constant up to second order in the initial angular velocity around the stable equi librium it is well known that for small oscillations of the pendulum and small intervals of time the dynamics of the pendulum can be ap proximated by the dynamics of the harmonic oscillator we study the dynamics of a family of mechanical systems that includes the pendulum at small neighbourhoods of the equilibrium but after long intervals of time so that the second order term of the period map can no longer be neglected we analyze such dynamical behaviour through a renor malization scheme acting on the dynamics of this family of mechanical systems the main theorem states that the asymptotic limit of this renormalization scheme is universal it is the same for all the elements in the considered class of mechanical systems as a consequence we obtain an universal asymptotic focal decomposition for this family of mechanical systems this paper is intended to be the first of a series of articles aiming at a semiclassical quantization of systems of the pendu lum type as a natural application of the focal decomposition associated to the two point boundary value problem

how fast can one compute the permanent of circulant matrices abstract in this paper we address the problem of computing the permanent of 0 1 circulant matrices we investigate structural properties of circulant matrices showing that i if they are dense enough then they contain large arbitrary submatrices and ii if they are very sparse then they are not too far from convertible matrices building upon ii we then develop an efficient algorithm which allows us to compute permanents of very sparse circulants of size up to 200

on the values of permanents of 0 1 circulant matrices with three ones per row abstract starting from previous works concerning permanents of 0 1 circulant matrices with three nonzero entries per row and just using algebraic methods we find some congruences satisfied by the values of such permanents along with simple generalizations to circulants with k ones per row k n k 3 we also give an alternative proof for known relations of fibonacci numbers with some of these values of permanents

invariants for smooth conjugacy of hyperbolic dynamical systems ii we show that the eigenvalues of the derivatives at periodic points form a complete set of invariants for smooth local conjugacy of anosov diffeomorphisms oft2

on the parity of permanents of circulant matrices abstract starting from previous results concerning determinants and permanents of 0 1 circulant matrices and using theory on finite fields with characteristic 2 we first see denoting by a the n n 0 1 circulant matrix whose 1 s in the first row are exactly in positions i 1 i 2 i k that per a is even iff gcd x i 1 x i 2 x i k 1 x n 1 in z 2 x we then derive some consequences for specific classes of primes n in particular when n is a prime such that the group z n is generated by the residue class 2 we see that for k n and k odd the value per a is always odd similarly for n 7 and n prime with the group of squares in z n generated by the class 2 we see that for k 3 the value per a is always odd contrary to such cases we find that when n is a mersenne prime greater than 3 there are a significant number of circulants a of the above form with k odd and per a even

differentiable conjugacy for groups of area preserving circle diffeomorphisms we study groups of circle diffeomorphisms whose action on the cylinder c s 1 s 1 preserves a volume form we first show that such a group is topologically conjugate to a subgroup of psl 2 r then discuss the existence of a differentiable conjugacy

hausdorff dimension bounds for smoothness of holonomies for codimension 1 hyperbolic dynamics we prove that the stable holonomies of a proper codimension 1 attractor lambda for a c r diffeomorphism f of a surface are not c1 theta for theta greater than the hausdorff dimension of the stable leaves of f intersected with lambda to prove this result we show that there are no diffeomorphisms of surfaces with a proper codimension 1 attractor that are affine on a neighbourhood of the attractor and have affine stable holonomies on the attractor

global invariants for measured foliations new exotic invariants for measured foliations are constructed which we call the u classes of a pair f u the dependence of the s classes on the geometry of the foliation gis examined and the dynamics of a foliation is shown to determine the u classes in many cases we use the u classes to study the classifying space bfsl of foliations with a transverse invariant volume form and we show the homotopy groups of bfsl are uncountably generated starting in degrees q 3 new invariants for groups of volume preserving diffeomorphisms also arise from the t classes these invariants are nontrivial and related to the geometric aspects of the group action relations between the tl classes and the secondary classes of a foliation are exhibited 0 introduction in this paper we construct new exotic characteristic classes for foliations which have a transverse invariant measure these invariants which we call the t classes are similar to the secondary classes and take values in the real cohomology h m of the foliated manifold m theorem 1 6 the values of the m classes are strongly related to the geometry of the foliation proposition 3 4 the u classes have applications to the study of foliations with a transverse invariant volume form theorems 5 1 and 5 2 to the computations of the homology groups of the discrete group of volume preserving diffeomorphisms of rw theorem 7 4 and to the problem of relating the secondary classes of a foliation with its global geometry 15 201 the a classes arose from an attempt to obtain general invariants of a foliation which are intermediate between the leaf and secondary classes they are a special case of a family of universal invariants associated to a c2 foliation the weil operators in codimension one the introduction of the weil operator is a decisive ingredient in duminy s elegant proof of the sullivan conjecture about the godbillon vey class 9 the properties of the weil operators in higher codimensions are given in 15 where they are used to study the residuable secondary classes we briefly describe the construction of the weil operators and their relations with the l classes and other foliation invariants in 2 for a foliation with all leaves compact we note that the weil operators are identically zero proposition 2 6 we denote by f a c2 foliation of codimension q on a paracompact manifold m and i denotes a transverse invariant measure for sf recall that ti assigns to each received by the editors june 28 1982 and in revised form november 9 1982 1980 mathematics subject classification primary 57r30 57r20 58f1 1 secondary 58h10 28d15

mean of the singularities of a gibbs measure we calculate the value of teh average of the singularities of a gibbs measure invariant with respect to an expansivec2 diffeomorphism of a one compact manifold this is the value related to dimension that one computes numerically we then define and study a function known as the correlation dimension which is related to a free energy function and we generalize the results in higher dimension with an axioma transformation acting on a two compact manifold

cantor exchange systems and renormalization we prove a one to one correspondence between i c1 conjugacy classes of c1 h cantor exchange systems that are c1 h fixed points of renormalization and ii c1 conjugacy classes of c1 h diffeomorphisms f with a codimension 1 hyperbolic attractor lambda that admit an invariant measure absolutely continuous with respect to the hausdorff measure on lambda however we prove that there is no c1 alpha cantor exchange system with bounded geometry that is a c1 alpha fixed point of renormalization with regularity alpha greater than the hausdorff dimension of its invariant cantor set

on invariant volumes of codimension one anosov flows and the verjovsky conjecture we show that any topologically transitive codimension one anosov flow on a closed manifold is topologically equivalent to a smooth anosov flow that preserves a smooth volume by a classical theorem due to verjovsky any higher dimensional codimension one anosov flow is topologically transitive recently simic showed that any higher dimensional codimension one anosov flow that preserves a smooth volume is topologically equivalent to the suspension of an anosov diffeomorphism therefore our result gives a complete classification of codimension one anosov flows up to topological equivalence in higher dimensions

resistance between two nodes in general position on an m n fan network the resistance between two nodes in general position on a fan network with n radial lines and m transverse lines is determined also a similar result of izmailian kenna and wu j phys a math theor 47 035003 2014 for an m ifmmode times else texttimes fi n cobweb network is reproduced but the method used here is significantly different it avoids the use of the kirchhoff matrix requires the solution of just one instead of two eigenvalue problems and results directly in only a single summation further the current distribution is given explicitly as a byproduct of the method the method is the same as that used by tan zhou and yang j phys a math theor 46 195202 2013 to find the cobweb resistance between center and perimeter for 1 ensuremath le m ensuremath le 3 and general n proof of their conjecture for general m is discussed

rigidity of some abelian by cyclic solvable group actions on mathbb t n in this paper we study a natural class of groups that act as affine transformations of mathbb t n we investigate whether these solvable abelian by cyclic groups can act smoothly and nonaffinely on mathbb t n while remaining homotopic to the affine actions in the affine actions elliptic and hyperbolic dynamics coexist forcing a priori complicated dynamics in nonaffine perturbations we first show using the kam method that any small and sufficiently smooth perturbation of such an affine action can be conjugated smoothly to an affine action provided certain diophantine conditions on the action are met in dimension two under natural dynamical hypotheses we get a complete classification of such actions namely any such group action by c r diffeomorphisms can be conjugated to the affine action by c r varepsilon conjugacy next we show that in any dimension c 1 small perturbations can be conjugated to an affine action via c 1 varepsilon conjugacy the method is a generalization of the herman theory for circle diffeomorphisms to higher dimensions in the presence of a foliation structure provided by the hyperbolic dynamics

the dynamics of open foliated manifolds and a vanishing theorem for the godbillon vey class let m f be a c3 foliated n manifold with leaves of dimension n 1 let gv f e h3 m ir denote the characeristic class introduced by godbillon and vey in g v considerable effort has been made to discover the geometric significance of this invariant recall thurston s helical wobble th which can be interpreted via the formulas in r w in addition to celebrated examples of foliations for which gv f does not vanish g v th br the list of foliations for which it does has been growing steadily by work of herman her and wallet wi gv sr 0 for foliated circle bundles and interval bundles over t2 this result has interesting consequences that are far from being obvious thus morita and tsuboi have deduced the vanishing of gv x for closed foliated manifolds without holonomy m t and mizutani morita and tsuboi have extended this to foliations almost without holonomy m m t in all of the above vanishing theorems the leaves of the foliations have polynomial growth other classes of foliations with all leaves polynomial and gv f 0 were found by nishimori n sergiescu ser and tsuchiya ts in a preprint c c2 that became obsolete within a few months of its distribution the present authors proved the vanishing theorem for all closed foliated manifolds in which every leaf has polynomial growth indeed we proved that if every leaf is nonresilient l and if no leaf lies at infinite level then gv st 0

bootstrap for local rigidity of anosov automorphisms on the 3 torus we establish a strong form of local rigidity for hyperbolic automorphisms of the 3 torus with real spectrum namely let l mathbb t 3 to mathbb t 3 be a hyperbolic automorphism of the 3 torus with real spectrum and let f be a c 1 small perturbation of l then f is smoothly c infty conjugate to l if and only if obstructions to c 1 conjugacy given by the eigenvalues at periodic points of f vanish by combining our result and a local rigidity result of kalinin and sadovskaya ergod theory dyn syst 29 117 136 2009 for conformal automorphisms this completes the local rigidity program for hyperbolic automorphisms in dimension 3 our work extends de la llave marco moriyon 2 dimensional local rigidity theory commun math phys 109 368 378 1987 ergod theory dyn syst 17 3 649 662 1997 commun math phys 109 4 681 689 1987

on a regularity problem occurring in connection with anosov diffeomorphisms let be ac manifold and s and u be two holder foliations transverse and with uniformlyc leaves if a functionf is uniformlyc along the leaves of the two foliations then it isc on the proof is elementary