recursion operators and non local symmetries a new formulation of recursion operators is presented which eliminates diffi culties associated with integro differential operators this interpretation treats recursion operators and their inverses on an equal footing efficient techniques for constructing non local symmetries of differential equations result

why nonlocal recursion operators produce local symmetries new results and applications it is well known that integrable hierarchies in 1 1 dimensions are local while the recursion operators that generate these hierarchies usually contain nonlocal terms we resolve this apparent discrepancy by providing simple and universal sufficient conditions for a nonlocal recursion operator in 1 1 dimensions to generate a hierarchy of local symmetries these conditions are satisfied by virtually all recursion operators known today and are much easier to verify than those found in earlier work we also give explicit formulae for the nonlocal parts of higher recursion hamiltonian and symplectic operators of integrable systems in 1 1 dimensions using these two results we prove under some natural assumptions the maltsev novikov conjecture stating that higher hamiltonian symplectic and recursion operators of integrable systems in 1 1 dimensions are weakly nonlocal i e the coefficients of these operators are local and these operators contain at most one integration operator in each term

divisibility of reduction in groups of rational numbers given a multiplicative group of nonzero rational numbers and a positive integer m we consider the problem of determining the density of the set of primes p for which the order of the reduction modulo p of the group is divisible by m in the case when the group is finitely generated the density is explicitly computed some examples of groups with infinite rank are considered

painleve classification of coupled korteweg de vries systems in this work we give a classification of coupled korteweg de vries equations we found new systems of equations that are completely integrable in the sense of painleve

two component integrable generalizations of burgers equations with nondiagonal linearity two component second and third order burgers type systems with nondiagonal constant matrix of leading order terms are classified for higher symmetries new integrable systems are obtained master symmetries of the obtained symmetry integrable systems and bi poisson structures of those that also possess conservation laws are given

sasa satsuma complex modified korteweg de vries ii and the complex sine gordon ii equation revisited recursion operators nonlocal symmetries and more we present a new symplectic structure and a hereditary recursion operator for the sasa satsuma equation which is widely used in nonlinear optics using an integrodifferential substitution relating this equation to a third order symmetry flow of the complex sine gordon ii equation enabled us to find a hereditary recursion operator and higher hamiltonian structures for the latter equation we also show that both the sasa satsuma equation and the third order symmetry flow for the complex sine gordon ii equation are bi hamiltonian systems and we construct several hierarchies of local and nonlocal symmetries for these systems

on recursion operators abstract we observe that application of a recursion operator of the burgers equation does not produce the expected symmetries this is explained by the incorrect assumption that d x 1 d x 1 we then proceed to give a method to compute the symmetries using the recursion operator as a first order approximation

lenard scheme for two dimensional periodic volterra chain we prove that for compatible weakly nonlocal hamiltonian and symplectic operators hierarchies of infinitely many commuting local symmetries and conservation laws can be generated under some easily verified conditions no matter whether the generating nijenhuis operators are weakly nonlocal or not we construct a recursion operator of the two dimensional periodic volterra chain from its lax representation and prove that it is a nijenhuis operator furthermore we show that this system is a generalized bi hamiltonian system rather surprisingly the product of its weakly nonlocal hamiltonian and symplectic operators gives rise to the square of the recursion operator

on euclidean ideal classes in certain abelian extensions in this article we show that certain abelian extensions k with unit rank greater than or equal to three have cyclic class group if and only if it has a euclidean ideal class this result improves an earlier result of murty and graves one can improve this result up to unit rank 2 if one assumes the elliott and halberstam conjecture see conjecture 1 in preliminaries these results are known under generalized riemann hypothesis by the work of lenstra j lond math soc 10 457 465 see also weinberger proc symp pure math 24 321 332

integrable systems and their recursion operators in this paper we discuss the structure of recursion operators we show that recursion operators of evolution equations have a nonlocal part that is determined by symmetries and cosymmetries this enables us to compute recursion operators more systematically under certain conditions which hold for all examples known to us nijenhuis operators are well defined i e they give rise to hierarchies of infinitely many commuting symmetries of the operator moreover the nonlocal part of a nijenhuis operator contains the candidates of roots and coroots

a strange recursion operator demystified we show that a new integrable two component system of kdv type studied by karasu kalkanli et al 2004 acta appl math 83 85 94 is bi hamiltonian and its recursion operator which has a highly unusual structure of nonlocal terms can be written as a ratio of two compatible hamiltonian operators found by us using this we prove that the system in question possesses an infinite hierarchy of local commuting generalized symmetries and conserved quantities in involution and the evolution systems corresponding to these symmetries are bi hamiltonian as well we also show that upon introduction of suitable nonlocal variables the nonlocal terms of the recursion operator under study can be written in the usual form with the integration operator d 1x appearing in each term at most once

on the r rank artin conjecture we assume the generalized riemann hypothesis and prove an asymptotic formula for the number of primes for which f p can be generated by r given multiplicatively independent numbers in the case when the r given numbers are primes we express the density as an euler product and apply this to a conjecture of brown zassenhaus j number theory 3 1971 306 309 finally in some examples we compare the densities approximated with the natural densities calculated with primes up to 9 10 4

primality and identity testing via chinese remaindering we give a simple and new randomized primality testing algorithm by reducing primality testing for number n to testing if a specific univariate identity over zn holds we also give new randomized algorithms for testing if a multivariate polynomial over a finite field or over rationals is identically zero the first of these algorithms also works over zn for any n the running time of the algorithms is polynomial in the size of arithmetic circuit representing the input polynomial and the error parameter these algorithms use fewer random bits and work for a larger class of polynomials than all the previously known methods for example the schwartz zippel test schwartz 1980 zippel 1979 chen kao and lewin vadhan tests chen and kao 1997 lewin and vadhan 1998

dispersionless 3 1 dimensional integrable hierarchies in this paper we introduce a multi dimensional version of the r matrix approach to the construction of integrable hierarchies applying this method to the case of the lie algebra of functions with respect to the contact bracket we construct integrable hierarchies of 3 1 dimensional dispersionless systems of the type recently introduced in sergyeyev 2014 http arxiv org abs 1401 2122

low order hamiltonian operators having momentum abstract we describe all fifth order hamiltonian operators in one dependent and one independent variable that possess momentum i e for which there exists a hamiltonian associated with translation in the independent variable similar results for first and third order hamiltonian operators were obtained earlier by mokhov

towards the complete classification of homogeneous two component integrable equations in this article we suggest an improved method for classifying general two component integrable evolution equations homogeneous in a given weighting scheme this method relies on linear changes of variables and on an appropriate splitting of the solution space to illustrate the method we implement the classification of coupled kdv type equations we show that there are five nontriangular systems possessing higher order generalized symmetries one of these systems is previously unknown this seems to be the first classification of coupled integrable equations homogeneous in a given weighting scheme without any restrictions on the form of the main matrix

oscillating solutions of the falkner skan equation for positive β on demontre l existence de solutions pour le probleme f ff 1 f 2 0 f o f o 0 f 1

the falkner skan equation ii dynamics and the bifurcations of p and q orbits the falkner skan equation is a reversible three dimensional system of ordinary differential equations with two distinguished straight line trajectories which form a heteroclinic loop between fixed points at infinity we showed in the previous paper 1995 j differential equations119 336 394 that at positive integer values of the parameter there are bifurcations creating large sets of periodic and other interesting trajectories here we show that all but two of these trajectories are destroyed in another sequence of bifurcations as and by considering topological invariants and orderings on certain manifolds we obtain unusually detailed information about the sequences of bifurcations which can occur

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

the falkner skan equation i the creation of strange invariant sets abstract we study bifurcations in the falkner skan equation we show that for positive integer values of the parameter there are bifurcations which create an infinity of periodic and other interesting trajectories these bifurcations provide a new mechanism for the creation of strange invariant sets in ordinary differential equations and are co dimension one

on a question of m newman on the number of commutators abstract m newman asked whether there is an absolute constant c such that every matrix in slnr is the product of at most c commutators where r ranges over euclidean commutative rings and n 3 we give here a negative answer however if for the ring r every matrix in slmr is the product of a bounded number of commutators for some fixed m 3 then for all sufficiently large n every matrix in slnr is the product of six commutators

computing prime harmonic sums 1801241484456448000 andthenusedaprecomputedtableofsumsintheschoenfeldinterval therebyavoiding interpolation or binary search and multiple evaluations of s x todiscovers 1801241230056600467 3 99999 99999 99999 99966 ands 1801241230056600523 4 00000 00000 00000 00021 there are no primes between these two xvalues this computation took roughly one week on two workstations franz mertensjosephine mitchell lowell schoenfeld

use of simple shooting to obtain chaos abstract we discuss an elementary method of proving that certain smooth differential equations exhibit a kind of chaos in some cases such as a forced pendulum the proof is simpler than an approach through poincare maps and melnikov theory in other cases new results are obtained even for the pendulum there is a generalization of previous results because the forcing need not be periodic or almost periodic

primes in intervals of bounded length in april 2013 yitang zhang proved the existence of a finite bound b such that there are infinitely many pairs of distinct primes which differ by no more than b this is a massive breakthrough makes the twin prime conjecture look highly plausible which can be re interpreted as the conjecture that one can take b 2 and his work helps us to better understand other delicate questions about prime numbers that had previously seemed intractable the original purpose of this talk was to discuss zhang s extraordinary work putting it in its context in analytic number theory and to sketch a proof of his theorem zhang had even proved the result with b 70 000 000 moreover a co operative team polymath8 collaborating only on line had been able to lower the value of b to 4680 not only had they been more careful in several difficult arguments in zhang s original paper they had also developed zhang s techniques to be both more powerful and to allow a much simpler proof indeed the proof of zhang s theorem that will be given in the write up of this talk is based on these developments in november inspired by zhang s extraordinary breakthrough james maynard dramatically slashed this bound to 600 by a substantially easier method both maynard and terry tao who had independently developed the same idea were able to extend their proofs to show that for any given integer m 1 there exists a bound bm such that there are infinitely many intervals of length bm containing at least m distinct primes we will also prove this much stronger result herein even showing that one can take bm e 8m if zhang s method is combined with the maynard tao set up then it appears that the bound can be further reduced to 576 if all of these techniques could be pushed to their limit then we would obtain b b2 12 so new ideas are still needed to have a feasible plan for proving the twin prime conjecture the article will be split into two parts the first half which appears here we will introduce the work of zhang polymath8 maynard and tao and explain their arguments that allow them to prove their spectacular results as we will discuss zhang s main novel contribution is an estimate for primes in relatively short arithmetic progressions the second half of this article sketches a proof of this result the bulletin article will contain full details of this extraordinary work part 1 primes in short intervals