edge decompositions of graphs with high minimum degree abstract a fundamental theorem of wilson states that for every graph f every sufficiently large f divisible clique has an f decomposition here a graph g is f divisible if e f divides e g and the greatest common divisor of the degrees of f divides the greatest common divisor of the degrees of g and g has an f decomposition if the edges of g can be covered by edge disjoint copies of f we extend this result to graphs g which are allowed to be far from complete in particular together with a result of dross our results imply that every sufficiently large k 3 divisible graph of minimum degree at least 9 n 10 o n has a k 3 decomposition this significantly improves previous results towards the long standing conjecture of nash williams that every sufficiently large k 3 divisible graph with minimum degree at least 3 n 4 has a k 3 decomposition we also obtain the asymptotically correct minimum degree thresholds of 2 n 3 o n for the existence of a c 4 decomposition and of n 2 o n for the existence of a c 2 l decomposition where l 3 our main contribution is a general iterative absorption method which turns an approximate or fractional decomposition into an exact one in particular our results imply that in order to prove an asymptotic version of nash williams conjecture it suffices to show that every k 3 divisible graph with minimum degree at least 3 n 4 o n has an approximate k 3 decomposition

lehmer s problem for polynomials with odd coefficients we prove that if f x ylkzo akxk is a polynomial with no cyclotomic factors whose coefficients satisfy ak 1 mod 2 for 0 2 we also characterize the polynomials that appear as the noncyclotomic part of a polynomial whose coefficients satisfy ak 1 mod p for each ft for a fixed prime p last we prove that the smallest pisot number whose minimal polynomial has odd coefficients is a limit point from both sides of salem 19 numbers whose minimal polynomials have coef ficients in 1 1

a more fundamental approach to the derivation of nonlinear acoustic wave equations with fractional loss operators l a corrected derivation of nonlinear wave propagation equations with fractional loss operators is presented the fundamental approach is based on fractional formulations of the stress strain and heat flux definitions but uses the energy equation and thermodynamic identities to link density and pressure instead of an erroneous fractional form of the entropy equation as done in prieur and holm nonlinear acoustic wave equations with fractional loss operators j acoust soc am 130 3 1125 1132 2011 the loss operator of the obtained nonlinear wave equations differs from the previous derivations as well as the dispersion equation but when approximating for low frequencies the expressions for the frequency dependent attenuation and velocity dispersion remain unchanged

the robust component structure of dense regular graphs and applications in this paper we study the large scale structure of dense regular graphs this involves the notion of robust expansion a recent concept which has already been used successfully to settle several longstanding problems roughly speaking a graph is robustly expanding if it still expands after the deletion of a small fraction of its vertices and edges our main result allows us to harness the useful consequences of robust expansion even if the graph itself is not a robust expander it states that every dense regular graph can be partitioned into robust components each of which is a robust expander or a bipartite robust expander we apply our result to obtain amongst others the following i we prove that whenever e 0 every sufficiently large 3 connected d regular graph on n vertices with d 1 4 e n is hamiltonian this asymptotically confirms the only remaining case of a conjecture raised independently by bollobas and haggkvist in the 1970s ii we prove an asymptotically best possible result on the circumference of dense regular graphs of given connectivity the 2 connected case of this was conjectured by bondy and proved by wei

fractal ladder models and power law wave equations extensive measurement of the ultrasonic attenuation coefficient in human and mammalian tissue has revealed a power law dependence on frequency to describe this power law behavior in tissue a hierarchical fractal network model is proposed the viscoelastic and self similar properties of tissue are captured by a constitutive equation based on a lumped parameter infinite ladder topology involving alternating springs and dashpots in the low frequency limit this ladder network yields a stress strain constitutive equation with a time fractional derivative by combining this constitutive equation with linearized conservation principles and an adiabatic equation of state the fractional partial differential equations previously proposed by m caputo linear models of dissipation whose q is almost frequency independent ii geophys j r astron soc 13 529 539 1967 and mg wismer finite element analysis of broadband acoustic pulses through inhomogeneous media with power law attenuation j acoust soc am 120 3493 3502 2006 are derived the resulting attenuation coefficient is a power law with exponents ranging between 1 and 2 while the phase velocity is in agreement with the kramers kronig relations the power law exponent is interpreted in terms of the mechanical structure and related to the spectral dimension of the underlying fractal geometry

mahler measure and entropy for commuting automorphisms of compact groups we compute the joint entropy ofd commuting automorphisms of a compact metrizable group letr d u 1 1 d 1 1 be the ring of laurent polynomials ind commuting variables andm be anr d module then the dual groupx m ofm is compact and multiplication onm by each of thed variables corresponds to an action m of d by automorphisms ofx m every action of d by automorphisms of a compact abelian group arises this way iff r d our main formula shows that the topological entropy of alpha r d left langle f right rangle is given by h left alpha r d left langle f right rangle right log m left f right int limits 0 1 ldots int limits 0 1 log left f left e 2 pi it 1 ldots e 2 pi it d right right dt 1 ldots dt d where m f is the mahler measure off this reduces to the classical result for toral automorphisms via jensen s formula while the entropy of a single automorphism of a compact group is always the logarithm of an algebraic integer this no longer seems to hold for joint entropy of commuting automorphisms since values such as 7 3 4 2 occur if p is a non principal prime ideal we show h left alpha r d p right 0 using an analogue of the yuzvinskii thomas addition formula we computeh m for arbitraryr d modulesm and then the joint entropy for an action of d on a not necessarily abelian compact group using a result of boyd we characterize those m which have completely positive entropy in terms of the prime ideals associated tom and show this condition implies that m is mixing of all orders we also establish an analogue of berg s theorem proving that if m has finite entropy then haar measure is the unique measure of maximal entropy if and only if m has completely positive entropy finally we show that for expansive actions the growth rate of the number of periodic points equals the topological entropy

new exact solutions of nonlinear fractional acoustic wave equations in ultrasound in this paper new exact solutions of fractional nonlinear acoustic wave equations have been devised the travelling periodic wave solutions of fractional burgers hopf equation and khokhlov zabolotskaya kuznetsov kzk equation have obtained by first integral method nonlinear ultrasound modelling is found to have an increasing number of applications in both medical and industrial areas where due to high pressure amplitudes the effects of nonlinear propagation are no longer negligible taking nonlinear effects into account the ultrasound beam analysis makes more accurate in these applications the burgers hopf equation is one of the extensively studied models in mathematical physics in addition the kzk parabolic nonlinear wave equation is one of the most widely employed nonlinear models for propagation of 3d diffraction sound beams in dissipative media in the present analysis these nonlinear equations have solved by first integral method as a result new exact analytical solutions have been obtained first time ever for these fractional order acoustic wave equations the obtained results are presented graphically to demonstrate the efficiency of this proposed method

entropy on abelian groups abstract we introduce the algebraic entropy for endomorphisms of arbitrary abelian groups appropriately modifying existing notions of entropy the basic properties of the algebraic entropy are given as well as various examples the main result of this paper is the addition theorem showing that the algebraic entropy is additive in appropriate sense with respect to invariant subgroups we give several applications of the addition theorem among them the uniqueness theorem for the algebraic entropy in the category of all abelian groups and their endomorphisms furthermore we point out the delicate connection of the algebraic entropy with the mahler measure and lehmer problem in number theory

hamilton decompositions of regular expanders a proof of kelly s conjecture for large tournaments abstract a long standing conjecture of kelly states that every regular tournament on n vertices can be decomposed into n 1 2 edge disjoint hamilton cycles we prove this conjecture for large n in fact we prove a far more general result based on our recent concept of robust expansion and a new method for decomposing graphs we show that every sufficiently large regular digraph g on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge disjoint hamilton cycles this enables us to obtain numerous further results e g as a special case we confirm a conjecture of erd s on packing hamilton cycles in random tournaments as corollaries to the main result we also obtain several results on packing hamilton cycles in undirected graphs giving e g the best known result on a conjecture of nash williams we also apply our result to solve a problem on the domination ratio of the asymmetric travelling salesman problem which was raised e g by glover and punnen as well as alon gutin and krivelevich

linking multiple relaxation power law attenuation and fractional wave equations the acoustic wave attenuation is described by an experimentally established frequency power law in a variety of complex media e g biological tissue polymers rocks and rubber recent papers present a variety of acoustical fractional derivative wave equations that have the ability to model power law attenuation on the other hand a multiple relaxation model is widely recognized as a physically based description of the acoustic loss mechanisms as developed by nachman et al j acoust soc am 88 1584 1595 1990 through assumption of a continuum of relaxation mechanisms each with an effective compressibility described by a distribution related to the mittag leffler function this paper shows that the wave equation corresponding to the multiple relaxation approach is identical to a given fractional derivative wave equation this work therefore provides a physically based motivation for use of fractional wave equations in acoustic modeling

relaxation matched modeling of propagation through porous media including fractal pore structure the viscous and thermal dissipation of an acoustic wave propagating through a porous medium is shown to be characteristic of a relaxation process based on this interpretation a new model for the complex density dynamic permeability and bulk modulus which describe the viscous and thermal processes respectively is proposed the model is based on approximating the relaxational characteristic as opposed to previous models based on matching low and high frequency asymptotic behavior an advantage of the relaxation model is that one fewer parameter is required the relaxation model is also simpler than the commonly used zwikker kosten attenborough or biot allard models in the sense that it contains no bessel or kelvin functions and is not a modified form of the solution for uniform circular pores and unlike the delany bazley empirical equations the relaxation model is physically realistic for all frequencies extension of the relaxation model to fractal pore surfaces is also discussed

justification for power laws and fractional models in previous chapters it has been shown that the standard linear viscoelastic models can be generalized to their fractional counterparts it has also been demonstrated that the fractional zener kelvin voigt maxwell and newtonian models all satisfy physical criteria such as passivity and causality these models have temporal responses that are characterized by power laws either exactly or asymptotically and this corresponds to observed temporal characteristics in many media the character of the fractional derivative operator is also such that it describes temporal memory in the media

dimension and entropy in compact topological groups abstract we study the topological entropy h f of continuous endomorphisms f of compact like groups more specifically we consider the e spectrum e t o p k for a compact like group k namely the set of all values h f when f runs over the set e n d k of all continuous endomorphisms of k we pay particular attention to the class e of topological groups without continuous endomorphisms of infinite entropy i e e t o p k as well as the subclass e 0 of e consisting of those groups k with e t o p k 0 it turns out that the properties of the e spectrum and these two classes are very closely related to the topological dimension we show among others that a compact connected group k with finite dimensional commutator subgroup belongs to e if and only if dim k and we obtain a simple formula involving the entropy function for the dimension of an abelian topological group which is either locally compact or bounded in particular compact examples are provided to show the necessity of the compactness or commutativity conditions imposed for the validity of these results e g compact connected semi simple groups k with dim k and k e 0 or countably compact connected abelian groups with the same property since the class e is not stable under taking closed subgroups or quotients we study also the largest subclasses s e and q e respectively of e having these stability properties we provide a complete description of these two classes in the case of compact groups that are either abelian or connected the counterpart for s e 0 and q e 0 is done as well

algebraic entropy for abelian groups the theory of endomorphism rings of algebraic structures allows in a natural way a systematic approach based on the notion of entropy borrowed from dynamical systems here we study the algebraic entropy of the endomorphisms of abelian groups introduced in 1965 by adler konheim and mcandrew the so called addition theorem is proved this expresses the algebraic entropy of an endomorphism of a torsion group as the sum of the algebraic entropies of the restriction to a invariant subgroup and of the endomorphism induced on the quotient group particular attention is paid to endomorphisms with zero algebraic entropy as well as to groups all of whose endomorphisms have zero algebraic entropy the significance of this class arises from the fact that any group not in this class can be shown to have endomorphisms of infinite algebraic entropy and we also investigate such groups a uniqueness theorem for the algebraic entropy of endomorphisms of torsion abelian groups is proved

endomorphisms of abelian groups with small algebraic entropy we study the endomorphisms of abelian groups g having a small algebraic entropy h where small usually means h log2 using essentially elementary tools from linear algebra we show that this study can be carried out in the group qd where an automorphism with h log2 must have all eigenvalues in the open circle of radius 2 centered at 0 and must leave invariant a lattice in qd i e be essentially an automorphism of zd in particular all eigenvalues of an automorphism with h 0 must be roots of unity this is a particular case of a more general fact known as algebraic yuzvinskii theorem we discuss other particular cases of this fact and we give some applications of our main results

hamilton decompositions of regular tournaments we show that every sufficiently large regular tournament can a lmost completely be decomposed into edge disjoint hamilton cycles more precisely for each 0 every regular tournament g of sufficiently large ordern contains at least 1 2 n edge disjoint hamilton cycles this gives an approximate solution to a conjecture of kelly from 1968 our result also extends to almost regular tournaments

hierarchical analogues to fractional relaxation equations recently fractional calculus has become an important tool in the analysis of slow relaxation phenomena such as stress strain relationships in polymeric materials in the rheological constitutive equations this implies the replacement of the first order derivatives by fractional derivatives here we show that such procedures have hierarchical mechanical analogues we focus on the generalized dashpot and the generalized maxwell model and display the corresponding arrangements our models allow a transparent interpretation of the parameters which enter the fractional equations and reveal that the internal dynamics are hierarchically constrained

algebraic entropy for amenable semigroup actions abstract we introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups s on discrete abelian groups a by endomorphisms they extend the classical algebraic entropy for endomorphisms of abelian groups corresponding to the case s n we investigate the fundamental properties of the algebraic entropy and compute it in several examples paying special attention to the case when s is an amenable group for actions of cancellative right amenable monoids on torsion abelian groups we prove the so called addition theorem in the same setting we see that a bridge theorem connects the algebraic entropy with the topological entropy of the dual action by means of pontryagin duality so that we derive an addition theorem for the topological entropy of actions of cancellative left amenable monoids on totally disconnected compact abelian groups

average value of solutions for the bipartite boolean quadratic programs and rounding algorithms we consider domination analysis of approximation algorithms for the bipartite boolean quadratic programming problem bbqp with m n variables a closed form formula is developed to compute the average objective function value a of all solutions in o m n time however computing the median objective function value of the solutions is shown to be np hard also we show that any solution with objective function value no worse than a dominates at least 2 m n 2 solutions and this bound is the best possible further we show that such a solution can be identified in o m n time and hence the domination ratio of this algorithm is at least 1 4 we then show that for any fixed natural numbers a and b such that a b 1 no polynomial time approximation algorithm exists for bbqp with domination ratio larger than 1 2 1 m n unless p np it is shown that some powerful local search algorithms can get trapped at a local maximum with objective function value less than a one of our approximation algorithms has an interesting rounding property which provides a data dependent lower bound on the optimal objective function value a new integer programming formulation of bbqp is also given and computational results with our rounding algorithms are reported

the bilinear assignment problem complexity and polynomially solvable special cases in this paper we study the bilinear assignment problem bap with size parameters m and n m le n bap is a generalization of the well known quadratic assignment problem and the three dimensional assignment problem and hence np hard we show that bap cannot be approximated within a constant factor unless p np even if the associated quadratic cost matrix q is diagonal further we show that bap remains np hard if m o root r of n for some fixed r but is solvable in polynomial time if m o left frac log n log log n right when the rank of q is fixed bap is observed to admit fptas and when this rank is one it is solvable in polynomial time under some additional restrictions we then provide a necessary and sufficient condition for bap to be equivalent to two linear assignment problems a closed form expression to compute the average of the objective function values of all solutions is presented whereas the median of the solution values cannot be identified in polynomial time unless p np we then provide polynomial time heuristic algorithms that find a solution with objective function value no worse than that of m 1 n 1 solutions however computing a solution whose objective function value is no worse than that of m n lceil frac m beta rceil lceil frac n beta rceil solutions is np hard for any fixed rational number beta 1

a causal and fractional all frequency wave equation for lossy media this work presents a lossy partial differential acoustic wave equation including fractional derivative terms it is derived from first principles of physics mass and momentum conservation and an equation of state given by the fractional zener stress strain constitutive relation for a derivative order in the fractional zener relation the resulting absorption k obeys frequency power laws as k 1 in a low frequency regime k 1 2 in an intermediate frequency regime and k 1 in a high frequency regime the value 1 corresponds to the case of a single relaxation process the wave equation is causal for all frequencies in addition the sound speed does not diverge as the frequency approaches infinity this is an improvement over a previously published wave equation building on the fractional kelvin voigt constitutive relation

deriving fractional acoustic wave equations from mechanical and thermal constitutive equations it is argued that fractional acoustic wave equations come in two kinds the first kind is constructed ad hoc to have loss operators that fit power law measurements the second kind is more fundamental as they in addition are based on underlying physical equations here that means constitutive equations these equations are the fractional kelvin voigt and the more general fractional zener stress strain relationships as well as a fractional version of the fourier heat law the properties of the wave equations are given in terms of attenuation and phase group velocities for low intermediate and high frequency regions in the most general case the attenuation exhibits power law behavior in all frequency ranges while the phase and group velocities increase sharply in the intermediate frequency range and converge to a constant finite value for high frequencies it is also shown that the fractional zener wave equation is equivalent to the multiple relaxation model for attenuation

decomposing tournaments into paths we consider a generalisation of kelly s conjecture which is due to alspach mason and pullman from 1976 kelly s conjecture states that every regular tournament has an edge decomposition into hamilton cycles and this was proved by kuhn and osthus for large tournaments the conjecture of alspach mason and pullman asks for the minimum number of paths needed in a path decomposition of a general tournament formula presented there is a natural lower bound for this number in terms of the degree sequence of formula presented and it is conjectured that this bound is correct for tournaments of even order almost all cases of the conjecture are open and we prove many of them

proof of a conjecture of thomassen on hamilton cycles in highly connected tournaments a conjecture of thomassen from 1982 states that for every k there is an f k s o that every strongly f k connected tournament contains k edge disjoint hamilton cycles a classical theorem of camion that every strongly connected tournament contains a hamilton cycle implies that f 1 1 so far even the existence of f 2 was open in this paper we prove thomassen s conjecture by showing that f k o k 2 log 2 k this is best possible up to the logarithmic factor as a tool we show that every strongly 10 4 k log k connected tournament is k linked which improves a previous exponential bound the proof of the latter is based on a fundamental result

a bandwidth theorem for approximate decompositions we provide a degree condition on a regular n vertex graph g which ensures the existence of a near optimal packing of any family h of bounded degree n vertex k chromatic separable graphs into g in general this degree condition is best possible here a graph is separable if it has a sublinear separator whose removal results in a set of components of sublinear size equivalently the separability condition can be replaced by that of having small bandwidth thus our result can be viewed as a version of the bandwidth theorem of bottcher schacht and taraz in the setting of approximate decompositions more precisely let k be the infimum over all 1 2 ensuring an approximate k k decomposition of any sufficiently large regular n vertex graph g of degree at least n now suppose that g is an n vertex graph which is close to r regular for some r k o 1 n and suppose that h 1 h t is a sequence of bounded degree n vertex k chromatic separable graphs with i e h i 1 o 1 e g we show that there is an edge disjoint packing of h 1 h t into g if the h i are bipartite then r 1 2 o 1 n is sufficient in particular this yields an approximate version of the tree packing conjecture in the setting of regular host graphs g of high degree similarly our result implies approximate versions of the oberwolfach problem the alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree

on the coefficient of nonlinearity β in nonlinear acoustics an analysis valid to second order in the acoustic variables is used to investigate the origin and form of the coefficient of nonlinearity two cases are considered first for progressive plane waves the well known formula is 1 b 2a where b 2a is the coefficient of the quadratic term in the isentropic pressure density relation it is shown that the component 1 which is associated with convection comes solely from nonlinear terms in the equation of continuity the second component comes from a quadratic term in the equation of state the momentum equation is linear for this type of wave motion and therefore contributes nothing to the expression for second for noncollinear interaction of two plane waves quasilinear analysis shows that the expression for appropriate for the sum and difference frequency waves has three components in this case the momentum equation does contribute to the formula for most practical problems the three component formula reduces to cos b 2a where is the angle of interaction the factor cos represents the combination of contributions cos2 2 and sin2 2 from the continuity and momentum equations respectively

note on matching preclusion number of random graphs abstract a perfect matching in a graph is a set of edges such that every vertex is incident with exactly one edge in this set the matching preclusion number of an even graph g denoted by m p g is the minimum number of edges whose deletion leaves the resulting graph without a perfect matching in this paper we study the matching preclusion number of random graphs and prove that if n is even and p log n log log n n then asymptotically almost surely m p g n p g n p